Appendix E: Using Binomial and Poisson Distributions

This appendix is intended to break through the fear often associated with the use of the binomial and Poisson probability distributions. It provides an introduction to the theory of probability and the mathematics involved in making decisions based upon these distributions. It also introduces easy-to-use charts to simplify calculations. Furthermore, this appendix illustrates the practical application of probability theory to quality control decision-making.

Introduction to discrete distributions

There are two basic types of distributions—continuous and discrete. Continuous distributions are used to make predictions based upon variables data. The most familiar continuous distribution is the "normal" distribution, characterized by the bellcurve. Discrete distributions are used to make predictions based upon attribute data. It is two of these discrete distributions that we will be dealing with in this appendix.

The terms probability, binomial distribution and Poisson distribution generally elicit fear in non-statisticians. However, the mystique surrounding these concepts shrouds some fairly simple and useful decision-making tools and concepts. By far the most important is probability, which may be defined as the ratio of the chances of an event occurring to the total number of possibilities. For example, in the simplest terms, a coin has two sides. When flipped it could land on either heads or tails. The probability of it landing with the head side up is 1/2 or 50 percent.

Before plunging headlong into the binomial and Poisson distributions, let us take a quick look at some very familiar applications of probability theory:

Example 1. What is the probability of drawing an ace from a regulation deck of 52 playing cards?

• 4/52 = 1/13 or 8 percent

Example 2. What is the probability of drawing two consecutive aces from that deck (assuming that the first ace is not returned to the deck)?

• 4/52 x 3/51 = 12/2,652 or 0.5 percent

These applications are fairly straightforward and unimposing. We intuitively know that the chances of drawing two consecutive aces are much lower than drawing the first ace. But simple mathematical calculations allow us to quantify those chances, or risks.

Binomial distribution

The binomial distribution is composed of independent trials having dichotomous outcomes and a constant probability of occurrence. For example, coin flipping. The mathematical formula defining the binomial distribution is:

• P (x) = (n!)/ x! (n - x)! (p^x q^n-x)

Where:

n	=	number of trials
x	=	number of occurrences
p	=	probability of occurrence
q	=	1-p

Example 3. Using coin-flipping, suppose we were to flip a coin 10 times. What is the probability of obtaining 3 heads?

n = 10	x = 3	p = 1/2 or 0.5	q = 1 -0.5 = 0.5
P(3) = (10!)/ 3!(10 - 3)! (.53) (.57) = 0.117 or 11.7 percent

That looks fairly simple when someone else is doing the calculating. However, the opportunities for errors are great and such calculations can become tedious. An easier approach is to use a set of precalculated tables—see the tables at the end of this appendix. For example, in the previous problem, simply find the section for the number of trials (n) and the number of occurrences (x). Read across this row to where it intersects with the column corresponding to the probability of occurrence (p). Read the value at that intersection -the value of 0.117. Very easy!

You will note that there is also a number in parentheses at the intersection. This is the cumulative probability of occurrence. In other words, this represents the probability of obtaining heads 3 or fewer times in the course of flipping the coin 10 times. This probability is 0.172.

Example 4. What would be the probability of drawing a heart on 4 consecutive draws, assuming that each card is returned to the deck after drawing?

n=4	x = 4	p = 13/52 = 0.25	q = 1 - .25 = .75
P(4) = 0.004 or 0.4 percent

Remember, the binomial distribution is generally applicable only in cases where there is a constant probability of occurrence. This is also known as the concept of sampling with replacement.

Poisson distribution

The Poisson distribution is similar to the binomial, except that it applies when the number of trials (n) is high, over 20, and/or the probability of occurrence (p) is small. It is defined by the mathematical expression:

• P(x) = ((np)^x e^-np)/ xl; where e = 2.71

Example 5. In our coin-flipping example, let's increase our number of flips to 20.

n = 20	x = 3	p = 0.5
P(3) = ((20 x .5)3 (2.71)-(20 x.5)) / 3! = 0.007 or 0.7 percent

Again, the calculations are not overly simplistic, rather they can be ominous and very difficult to calculate. Therefore, a set of Poisson tables have been developed to facilitate making such predictions. Using the same problem, first we must calculate an "np" value. This simply involves multiplying "n" times "p" - (20 0.5) = 10.0. We find the column under the "np" of 10.0 and read across at the row for our "x" (3). These intersect at 0.007 or 0.7 percent. The reader will also note that we have provided a number in parentheses, which is the cumulative probability of occurrence. Therefore, the probability of 3 or fewer heads in 20 flips is 0.009 or 0.9 percent.

Example 6. Going back to our friendly deck of cards, what would be the probability of drawing four aces in 13 draws?

n = 13	x = 4	p = 4/52	np = 13 4/52 = 1.0
P(4) = 0.016 or 1.6 percent

Now that we know how to use these two charts, how do we know when to use each chart? The following rule is applicable. Start with the binomial chart. If the exact "n" and "p" values are not there, then calculate "np" and go to the Poisson chart. Let us review this process by solving the next couple of a few problems (#7 and #8).

Example 7. If, on average, a new product produced by a particular company has six defects, what is the probability that the new product you bought from them will have no defects?

n = 1	x= 0	p = 6.0
Poisson, np = 6.0
P(0) = 0.002 or 0.2 percent

Example 8. Suppose that a supplier sends you a lot of material that you find to be 10% defective, but he claims passed his final inspection sampling calling for a sample size of 25 and a reject number of 2. What is the probability that his final inspector properly sampled this lot?

n = 25	x = 2	p = 0.10
Poisson, np = 2.5
P(2) = 0.287 or 28.7 percent

In other words, there is nearly a 3 in 4 chance that the final inspection process was flawed.

Conclusion

Despite their forbidding formulae, the binomial and Poisson probability distributions are not difficult to use. They do have limitations. As with any attribute inspection system, they are limited in their application. However, the reader will notice that the potential for these types of examples is unlimited. In fact, these discrete distributions can be valuable, if properly used, in establishing sampling programs such as operating curves (OC) which are developed from the discrete probability formulas, in life testing approximations, or in evaluating the effectiveness of sampling programs. As in Example 8, the results can lead to potential problem areas. The binomial and Poisson distributions are merely two more tools in the toolbox of a professional who deals in quality issues. Too often they have been ignored because we didn't know how to use them. That should never be the case again.

EXAMPLE 3

n	x	.40	.45	.50
10	0	.006(0.006)	.002(0.002)	.001(0.001)
	1	.040(0.046)	.021(0.023)	.010(0.011)
	2	.121(0.167)	.076(0.099)	.044(0.055)
	3	.215(0.382)	.167(0.266)	.117(0.172)
	4	.251(0.633)	.238(0.504)	.205(0.377)
	5	.201(0.834)	.234(0.738)	.246(0.623)
	.	.	.	.
	.	.	.	.
	.	.	.	.
	10	.000(1.000)	.000(1.000)	.001(1.000)

EXAMPLE 4

n	x	.15	.20	.25
4	0	.522(0.522)		.410(0.410)	.316(0.316)
	1	.368(0.890)		.410(0.820)	.422(0.738)
	3	.011 (0.999)		.025(0.998)	.047(0.996)
	4	.001(1.000)		.002(1.000)	.004(1.000)

EXAMPLE 5

np	8.0	9.0	10.0
x
0
1	.003(0.003)		.001(0.001)
3	.011(0.014)		.005(0.006)	.002(0.002)
4	.029(0.043)		.015(0.021)	.007(0.009)
2	.000(1.000)		.000(1.000)	.001(1.000)

EXAMPLE 6

np	0.8	0.9	1.0
c
0	.449(0,449)
1	.359(0.808)		.406(0.406)	.368(0.368)
2	.144(0.952)		.366(0.772)	.368(0.736)
3	.039(0.991)		.166(0.938)	.184(0.920)
4	.008 0.999		.049 0.987	.061 0.981
5			.011(0.998)	.016(0.997)
	.001(1.000)		.002(1.000)	.003(1.000)

EXAMPLE 7

np	6.0	7.0	8.0
c
0	.002(0.002)		.001(0.001)
1	.015(0.017)		.006(0.007)	.003(0.003)
2	.045(0.062)		.022(0.29)	.011(0.014)
.	.	.	.
.	.	.	.
.	.	.	.
18	.000(1.000)		.000(1.000)	.001(1.000)

EXAMPLE 8

np	2.3	2.4	2.5
c
0	.100(0.100)		0.91(0.091)	.082(0.082)
1	.231(0.331)		.218(0.309)	.205(0.287)
2	.265(0.596)		.261(0.570)	.256(0.543)
3	.203(0.799)		.209(0.779)	.214(0.757)
.	.	.	.
.	.	.	.
.	.	.	.
9	.000(1.000)		.001(1.000)	.001(1.000)

Table E.1: Poisson probability distribution

np
c	0.1	0.2	0.3	0.4	0.5
0	.905(0.905)	.819(0.819)	.741(0.741)	.670(0.670)	.607(0.607)
1	.09.1(0.996)	.164(0.983)	.222(0.963)	.268(0.938)	.303(0.910)
2	.004(1.000)	.016(0.999)	.033(0.996)	.054(0.992)	.076(0.986)
3		.010(1.000)	.004(1.000)	.007(0.999)	.013(0.999)
4				.001(1.000)	.001(1.000)
np
c	0.6	0.7	0.8	0.9	1.0
0	.549(0.549)	.497(0.497)	.449(0.449)	.406(0.406)	.368(0.368)
1	.329(0.878)	.349(0.845)	.359(0.808)	.366(0.772)	.368(0.736)
2	.099(0.977)	.122(0.967)	.144(0.952)	.166(0.938)	.184(0.920)
3	.020(0.997)	.028(0.995)	.039(0.991)	.049(0.987)	.061(0.981)
4	.003(1.000)	.005(1.000)	.008(0.999)	.011(0.998)	.016(0.997)
5			.001(1.000)	.002(1.000)	.003(1.000)
np
c	1.1	1.2	1.3	1.4	1.5
0	.333(0.333)	.301(0.301)	.273(0.273)	.247(0.247)	.223(0.223)
1	.366(0.699)	.361(0.662)	.354(0.627)	.345(0.592)	.335(0.558)
2	.201(0.900)	.217(0.879)	.230(0.857)	.242(0.834)	.251(0.809)
3	.074(0.974)	.087(0.966)	.100(0.957)	.113(0.947)	.126(0.935)
4	.021(0.995)	.026(0.992)	.032(0.989)	.039(0.986)	.047(0.982)
5	.004(0.999)	.007(0.999)	.009(0.998)	.011(0.997)	.014(0.996)
6,	.001(1.000)	.001(1.000)	.002(1.000)	.003(1.000)	.004(1.000)
np
c	1.6	1.7	1.8	1,9	2.0
0	.202(0.202)	.183(0.183)	.165(0.165)	.150(0.150)	.135(0.135)
1	.323(0.525)	.311(0.494)	.298(0.463)	.284(0.434)	.271(0.406)
2	.258(0.783)	.264(0.758)	.268(0.731)	.270(0.704)	.271(0.677)
3	.138(0.921)	.149(0.907)	.161(0.892)	.171(0.875)	.180(0.857)
4	.055(0.976)	.064(0.971)	.072(0.964)	.081(0.956)	.090(0.947)
5	.018(0.994)	.022(0.993)	.026(0.990)	.031(0.987)	.036(0.983)
6	.005(0.999)	.006(0.999)	.008(0.998)	.010(0.997)	.012(0.995)
7	.001(1.000)	.001(1.000)	.002(1.000)	.003(1.000)	.004(0.999)
8					.001(1.000)
np
c	2.1	2.2	2.3	2.4	2.5
0	.123(0.123)	.111(0.111)	.100(0.100)	.091(0.091)	.082(0.082)
1	.257(0.380)	.244(0.355)	.231(0.331)	.218(0.309)	.205(0.287)
2	.270(0.650)	.268(0.623)	.265(0.596)	.261(0.570)	.256(0.543)
3	.189(0.839)	.197(0.820)	.203(0.799)	.209(0.779)	.214(0.757)
4	.099(0.938)	.108(0.928)	.117(0.916)	.125(0.904)	.134(0.891)
5	.042(0.980)	.048(0.976)	.054(0.970)	.050(0.964)	.067(0.958)
6	.015(0.995)	.017(0.993)	.021(0.991)	.024(0.988)	.028(0.986)
7	.004(0.999)	.005(0.998)	.007(0.998)	.008(0.996)	.010(0.996)
8	.001(1.000)	.002(1.000)	.002(1.000)	.003(0.999)	.003(0.999)
9				.001(1.000)	.001(1.000)
np
c	2.6	2.7	2.8	2.9	3.0
0	.074(0.074)	.067(0.067)	.061(0.061)	.055(0.055)	.050(0.050)
1	.193(0.267)	.182(0.249)	,170(0.231)	.160(0.215)	.149(0.199)
2	.251(0.518)	.245(0.494)	.238(0.469)	.231(0.446)	.224(0.423)
3	.218(0.736)	.221(0.715)	.223(0.692)	.224(0.670)	.224(0.647)
4	.141(0.877)	.149(0.864)	.156(0.848)	.162(0.832)	.168(0.815)
5	.074(0.951)	.080(0.944)	.087(0.935)	.094(0.926)	.101(0.916)
6	.032(0.983)	.036(0.980)	.041(0.976)	.045(0.971)	.050(0.966)
7	.012(0.995)	.014(0.994)	.016(0.992)	.019(0.990)	.022(0.988)
8	.004(0.999)	.005(0.999)	.006(0.998)	.007(0.997)	.008(0.996)
9	.001(1.000)	.001(1.000)	.002(1.000)	.002(0.999)	.003(0.999)
10				.001(1.000)	.001(1.000)
np
c	3.1	3.2	3.3	3.4	3.5
0	.045(0.045)	.041(0,041)	.037(0.037)	.033(0.033)	.030(0.030)
1	.140(0.185)	.130(0.171)	.122(0.159)	.113(0.146)	.106(0.136)
2	.216(0.401)	.209(0.380)	.201(0.360)	.193(0.339)	.185(0.321)
3	.224(0.625)	.223(0.603)	.222(0.582)	.219(0.558)	.216(0.537)
4	.173(0.798)	.178(0.781)	.182(0.764)	.186(0.744)	.189(0.726)
5	.107(0.905)	.114(0.895)	.120(0.884)	.126(0.870)	.132(0.858)
6	.056(0.961)	.061(0.956)	.066(0.950)	.071(0.941)	.077(0.935)
7	.025(0.986)	.028(0.984)	.031(0.981)	.035(0.976)	.038(0.973)
8	.010(0.996)	.011(0.995)	.012(0.993)	.015(0.991)	.017(0.990)
9	.003(0.999)	.004(0.999)	.005(0.998)	.006(0.997)	.007(0.997)
10	.001(1.000)	.001(1.000)	.002(1.000)	.002(0.999)	.002(0.999)
11				.001(1.000)	.001(1.000)
np
c	3.6	3.7	3.8	3.9	4.0
0	.027(0.027)	.025(0.025)	.022(0.022)	.020(0.020)	.018(0.018)
1	.098(0.125)	.091(0.116)	.085(0.107)	.079(0.099)	.073(0.091)
2	.177(0.302)	.169(0.285)	.161(0.268)	.154(0.253)	.147(0.238)
	.213(0.515)	,209(0.494)	.205(0.473)	.200(0.453)	.195(0.433)
4	.191(0.706)	.193(0.68'7)	.194(0.667)	.195(0.648)	.195(0.628)
5	.138(0.844)	.143(0.830)	.148(0.815)	.152(0.800)	.157(0.785)
6	.083(0.927)	.088(0.918)	.094(0.909)	.099(0.899)	.104(0.889)
7	.042(0.969)	.047(0.965)	.051(0.960)	.055(0.954)	.060(0.949)
8	.019(0.088)	.022(0.987)	.024(0.984)	.027(0.981)	.030(0.979)
9	.008(0.996)	.009(0.996)	.010(0.994)	.012(0.993)	.013(0.992)
10	.003(0.999)	.003(0.999)	.004(0.998)	.004(0.997)	.005(0.997)
11	.001(1.000)	.001(1.000)	.001(0.999)	.002(0.999)	.002(0.999)
12			.001(1.000)	.001(1.000)	.001(1.000)
np
c	4.1	4.2	4.3	4.4	4.5
0	.017(0.017)	.015(0.015)	.014(0.014)	.012(0.012)	.011(0.011)
1	.068(0.085)	.063(0.078)	.058(0.072)	.054(0.066)	.050(0.061)
2	.139(0.224)	.132(0.210)	.126(0.198)	.119(0.185)	.113(0.174)
3	.190(0.414)	.185(0.395)	.180(0.378)	.174(0.359)	.169(0.343)
4	.195(0.609)	.195(0.590)	.193(0.571)	.192(0.551)	.190(0.533)
5	.160(0.769)	.163(0.753)	.166(0.737)	.169(0.720)	.171(0.704)
6	.110(0.879)	.114(0.867)	.119(0,856)	.124(0.844)	.128(0.832)
7	.064(0.943)	.069(0.936)	.073(0.929)	.078(0.922)	.082(0.914)
8	.033(0.976)	.036(0.972)	.040(0.969)	.043(0.965)	.046(0.960)
9	.015(0.991)	.017(0.989)	.019(0.988)	.021(0.986)	.023(0.983)
10	.006(0.997)	.007(0.996)	.008(0.996)	.009(0.995)	.011(0.994)
11	.002(0.999)	.003(0.999)	.003(0.999)	.004(0.999)	.004(0.998)
12	.001(1.000)	.001(1.000)	.001(1.000)	.001(1.000)	.001(0.999)
13					.001(1.000)
np
c	4.6	4.7	4.8	4.9	5.0
0	.010(0.010)	.009(0.009)	.008(0.008)	.008(0.008)	.007(0.007)
1	.046(0.056)	.043(0.052)	.039(0.047)	.037(0.045)	.034(0.041)
2	.106(0.162)	.101(0.153)	.095(0.142)	.090(0.135)	.084(0.125)
3	.163(0.325)	.157(0.310)	.152(0.294)	.146(0.281)	.140(0.265)
4	.188(0.513)	.185(0.495)	.182(0.476)	.179(0.460)	.176(0.441)
5	.172(0.685)	.174(0.669)	.175(0.651)	.175(0.625)	.176(0.617)
6	.132(0.817)	.136(0.805)	.140(0.791)	.143(0.778)	.146(0.763)
7	.087(0.904)	.091(0.896)	.096(0.887)	.100(0.878)	.105(0.868)
8	.050(0.954)	.054(0.950)	.058(0.945)	.061(0.939)	.065(0.933)
9	.026(0.980)	.028(0.978)	.031(0.976)	.034(0.973)	.036(0.969)
10	.012(0.992)	.013(0.991)	.015(0.99.1)	.016(0.989)	.018(0.987)
11	.005(0.997)	.006(0.997)	.006(0.997)	.007(0.996)	.008(0.995)
12	.002(0.999)	.002(0.999)	.002(0.999)	.003(0.999)	.003(0.998)
13	.001(1.000)	.001(1.000)	.001(1.000)	.001(1.000)	.001(0.999)
14					.001(1.000)
np
c	6.0	7.0	8.0	9.0	10.0
0	.002(0.002)	.001(0.001)
1	.015(0.017)	.006(0.007)	.003(0.003)	.001(0.001)
2	.045(0.062)	.022(0.029)	.011(0.014)	.005(0.006)	.002(0.002)
3	.089(0.151)	.052(0.081)	.029(0.043)	.015(0.021)	.007(0.009)
4	.134(0.285)	.091(0.172)	.057(0.100)	.034(0.055)	.019(0.028)
5	.161(0.446)	.128(0.300)	.092(0.192)	.061(0.116)	.038(0.066)
6	.161(0.607)	.149(0.449)	.122(0.314)	.091(0.207)	.063(0.129)
7	.138(0.745)	.149(0.598)	.140(0.454)	.117(0.324)	.090(0.219)
8	.103(0.848)	.131(0.729)	.140(0.594)	.132(0.456)	.113(0.332)
9	.069(0.917)	.102(0.831)	.124(0.718)	.132(0.588)	.125(0.457)
10	.041(0.958)	.071(0.902)	.099(0.817)	.119(0.707)	.125(0.582)
11	.023(0.981)	.045(0.947)	.072(0.889)	.097(0.804)	.114(0.696)
12	.011(0.992)	.026(0.973)	.048(0.937)	.073(0.877)	.095(0.791)
13	.005(0.997)	.014(0.987)	.030(0.967)	.050(0.927)	.073(0.864)
14	.002(0.999)	.007(0.994)	.017(0.984)	.032(0.959)	.052(0.916)
15	.001(1.000)	.003(0.997)	.009(0.993)	.019(0.978)	.035(0.951)
16		.002(0.999)	.004(0.997)	.011(0.989)	.022(0.973)
17		.001(1.000)	.002(0.999)	.006(0.995)	.013(0.986)
18			.001(1.000)	.003(0.998)	.007(0.993)
19				.001(0.999)	.004(0.997)
20				.001(1.000)	.002(0.999)
21					.001(1.000)
np
c	11.0	12.0	13.0	14.0	15.0
0
1
2	.001(0.001)
3	.004(0.005)	.002(0.002)	.001(0.001)
4	.010(0.015)	.005(0.007)	.003(0.004)	.001(0.001)	.001(0.001)
5	.022(0.037)	.013(0.020)	.007(0.011)	.004(0.005)	.002(0.003)
6	.041(0.078)	.025(0.045)	.015(0.026)	.009(0.014)	.005(0.008)
7	.065(0.143)	.044(0.089)	.028(0.054)	.017(0.031)	.010(0.018)
8	.089(0.232)	.066(0.155)	.046(0.100)	.031(0.062)	.019(0.037)
9	.109(0.341)	.087(0.242)	.066(0.166)	.047(0.109)	.032(0.069)
10	.119(0.460)	.105(0.347)	.086(0.252)	.066(0.175)	.049(0.118)
11	.119(0.579)	.114(0.461)	.101(0.353)	.084(0.259)	.066(0.184)
12	.109(0.688)	.114(0.575)	.110(0.463)	.099(0.358)	.083(0.267)
13	.093(0.781)	.106(0.681)	.110(0.573)	.106(0.464)	.096(0.363)
14	.073(0.854)	.091(0.772)	.102(0.675)	.106(0.570)	.102(0.465)
15	.053(0.907)	.072(0.844)	.088(0.763)	.099(0.669)	.102(0.567)
16	.037(0.944)	.054(0.898)	.072(0.835)	.087(0.756)	.096(0.663)
17	.024(0.963)	.038(0.936)	.055(0.890)	.071(0.827)	.085(0.748)
18	.015(0.983)	.026(0.962)	.040(0.930)	.056(0.883)	.071(0.819)
19	.008(0.991)	.016(0.973)	.027(0.957)	.041(0.924)	.056(0.875)
20	.005(0.996)	.010(0.988)	.018(0.975)	.029(0.953)	.042(0.917)
21	.002(0.998)	.006(0.994)	.011(0.986)	.019(0.972)	.030(0.947)
22	.001(0.999)	.003(0.997)	.006(0.992)	.012(0.984)	.020(0.967)
23	.001(1.000)	.002(0.999)	.004(0.996)	.007(0.991)	.013(0.980)
24		.001(.1.000)	.002(0.998)	.004(0.995)	.008(0.988)
25			.001(0.999)	.003(0.998)	.005(0.993)
26			.001(1.000)	.001(0.999)	.003(0.996)
27				.001(1.000)	.002(0.998)
28					.001(0.999)

Table E.2: Binomial probability distribution

n	x	0.05	0.10	0.15	0.20	0.25
1	0	0.950(0.950)	0.900(0.900)	0.850(0.850)	0.800(0.800)	0.750(0.750)
	1	0,050(1.000)	0.100(1.000)	0.150(1.000)	0.200(1.000)	0.250(1.000)
2	0	0.902(0.902)	0.810(0.810)	0.722(0.722)	0.640(0.640)	0.562(0.562)
	1	0.095(0.997)	0.180(0.990)	0.255(0.977)	0.320(0.960)	0.375(0.938)
	2	0.003(1.000)	0.010(1.000)	0.022(1.000)	0.040(1.000)	0.062(1.000)
3	0	0.857(0.857)	0.729(0.729)	0.614(0.614)	0.512(0.512)	0.422(0.422)
	1	0.135(0.993)	0.243(0.972)	0.325(0.939)	0.384(0.896)	0.422(0.844)
	2	0.007(1.000)	0.027(0.999)	0.057(0.997)	0.096(0.992)	0.141(0.984)
	3		0.001(1.000)	0.003(1.000)	0.008(1.000)	0.016(1.000)
4	0	0.815(0.815)	0.656(0.656)	0.522(0.522)	0.410(0.410)	0.316(0.316)
	1	0.171(0.986)	0.292(0.948)	0.368(0.890)	0.410(0.819)	0.422(0.738)
	2	0.014(1.000)	0.049(0.996)	0.098(0.988)	0.154(0.973)	0.211(0.949)
	3		0.004(1.000)	0.011(0.999)	0.026(0.998)	0.047(0.996)
	4			0.001(1.000)	0.002(1.000)	0.004(1.000)
5	0	0.774(0.774)	0.590(0.590)	0.444(0.444)	0.328(0.328)	0.237(0.237)
	1	0.204(0.977)	0.328(0.919)	0.392(0.835)	0.410(0.737)	0.396(0.633)
	2	0.021(0.999)	0.073(0.991)	0.138(0.973)	0.205(0.942)	0.264(0.896)
	3	0.001(1.000)	0.008(1.000)	0.024(0.998)	0.051(0.993)	0.088(0.984)
	4			0.002(1.000)	0.006(1.000)	0.015(0.999)
	5					0.001(1.000)
6	0	0.735(0.735)	0.531(0.531)	0.377(0.377)	0.262(0.262)	0.178(0.178)
	1	0.232(0.967)	0.354(0.886)	0.399(0.776)	0.393(0.655)	0.356(0.534)
	2	0.031(0.998)	0.098(0.984)	0.176(0.953)	0.246(0.901)	0.297(0.831)
	3	0.002(1.000)	0.015(0.999)	0.041(0.994)	0.082(0.983)	0.132(0.962)
	4		0.001(1.000)	0.005(1.000)	0.015(0.998)	0.033(0.995)
	5				0.002(1.000)	0.004(1.000)
	6
7	0	0.698(0.698)	0.478(0.478)	0.321(0.321)	0.210(0.210)	0.133(0.133)
	1	0.257(0.956)	0.372(0.850)	0.396(0.717)	0.367(0.577)	0.311(0.445)
	2	0.041(0.996)	0.124(0.974)	0.210(0.926)	0.275(0.852)	0.311(0.756)
	3	0.004(1.000)	0.023(0.997)	0.062(0.988)	0.115(0.967)	0.173(0.929)
	4		0.003(1.000)	0.011(0.999)	0.029(0.995)	0.058(0.987)
	5			0.001(1.000)	0.004(1.000)	0.012(0.999)
	6					0.001(1.000)
	7
8	0	0.663(0.663)	0.430(0.430)	0.272(0.272)	0.168(0.168)	0.100(0.100)
	1	0.279(0.943)	0.383(0.813)	0.385(0.657)	0.336(0.503)	0.267(0.367)
	2	0.051(0.994)	0.149(0.962)	0.238(0.895)	0.294(0.797)	0.311(0.679)
	3	0.005(1.000)	0.033(0.995)	0.084(0.979)	0.147(0.944)	0.208(0.886)
	4		0.005(1.000)	0.018(0,997)	0.046(0.990)	0.087(0.973)
	5			0.003(1.000)	0.009(0.999)	0.023(0.996)
	6				0.001(1.000)	0.004(1.000)
	7
	8
9	0	0.630(0.630)	0.387(0.387)	0.232(0.232)	0.134(0.134)	0.075(0.075)
	1	0.299(0.929)	0.387(0.775)	0.368(0.599)	0.302(0.436)	0.225(0.300)
	2	0.063(0.992)	0.172(0.947)	0.260(0.859)	0.302(0.738)	0.300(0.601)
	3	0.008(0.999)	0.045(0.992)	0.107(0.966)	0.176(0.914)	0.234(0.834)
	4	0.001(1.000)	0.007(0.999)	0.028(0.994)	0.066(0.980)	0.117(0.951)
	5		0.001(1.000)	0.005(0.999)	0.017(0.997)	0.039(0.990)
	6			0.001(1.000)	0.003(1.000)	0.009(0.999)
	7					0.001(1.000)
	8
	9
10	0	0.599(0.599)	0.349(0.349)	0.197(0.197)	0.107(0.107)	0.056(0.056)
	1	0.315(0.914)	0.387(0.736)	0.347(0.544)	0.268(0.376)	0.188(0.244)
	2	0.075(0.988)	0.194(0.930)	0.276(0.820)	0.302(0.678)	0.282(0.526)
	3	0.010(0.999)	0.057(0.987)	0.130(0.950)	0.201(0.879)	0.250(0.776)
	4	0.001(1.000)	0.011(0.998)	0.040(0.990)	0.088(0.967)	0.146(0.922)
	5		0.001(1.000)	0.008(0.999)	0.026(0.994)	0.058(0.980)
	6			0.001(1.000)	0.006(0.999)	0.016(0.996)
	7				0.001(1.000)	0.003(1.000)
	8
	9
	10
11	0	0.569(0.569)	0.314(0.314)	0.167(0.167)	0.086(0.086)	0.042(0.042)
	1	0.329(0.898)	0.384(0.697)	0.325(0.492)	0.236(0.322)	0.155(0.197)
	2	0.087(0.985)	0.213(0.910)	0.287(0.779)	0.295(0.617)	0.258(0.455)
	3	0.014(0.998)	0.071(0.981)	0.152(0.931)	0.221(0.839)	0.258(0.713)
	4	0.001(1.000)	0.016(0.997)	0.054(0.984)	0.111(0.950)	0.172(0.885)
	5		0.002(1.000)	0.013(0.997)	0.039(0.988)	0.080(0.966)
	6			0.002(1.000)	0.010(0.998)	0.027(0.992)
	7				0.002(1.000)	0.006(0.999)
	8					0.001(1.000)
	9
	10
	11
12	0	0.540(0.540)	0.282(0.282)	0.142(0.142)	0.069(0.069)	0.032(0.032)
	1	0.341(0.882)	0.377(0.659)	0.301(0.443)	0.206(0.275)	0.127(0.158)
	2	0.099(0.980)	0.230(0.889)	0.292(0.736)	0.283(0.558)	0.232(0.391)
	3	0.017(0.998)	0.085(0.974)	0.172(0.908)	0.236(0.795)	0.258(0.649)
	4	0.002(0.999)	0.021(0.996)	0.068(0.976)	0.133(0.927)	0.194(0.842)
	5	0.001(1.000)	0.004(0.999)	0.019(0.995)	0.053(0.981)	0.103(0.946)
	6		0.001(1.000)	0.004(0.999)	0.016(0.996)	0.040(0.986)
	7			0.001(1.000)	0.003(0.999)	0.011(0.997)
	8				0.001(1.000)	0.002(1.000)
	9
	10
	11
	12
13	0	0.513(0.513)	0.254(0.254)	0.121(0.121)	0.055(0.055)	0.024(0.024)
	1	0.351(0.865)	0.367(0.621)	0.277(0.398)	0.179(0.234)	0.103(0.127)
	2	0.111(0.975)	0.245(0.866)	0.294(0.692)	0.268(0.502)	0.206(0.333)
	3	0.021(0.997)	0.100(0.966)	0.190(0.882)	0.246(0.747)	0.252(0.584)
	4	0.003(1.000)	0.028(0.994)	0.084(0.966)	0.154(0.901)	0.210(0.794)
	5		0.006(0.999)	0.027(0.992)	0.069(0.970)	0.126(0.920)
	6		0.001(1.000)	0.006(0.999)	0.023(0.993)	0.056(0.976)
	7			0.001(1.000)	0.006(0.999)	0.019(0.994)
	8				0.001(1.000)	0.005(0.999)
	9					0.001(1.000)
	10
	11
	12
	13
14	0	0.488(0.488)	0.229(0.229)	0.103(0.103)	0.044(0.044)	0.018(0.018)
	1	0.359(0.847)	0.356(0.585)	0.254(0.357)	0.154(0.198)	0.083(0.101)
	2	0.123(0.970)	0.257(0.842)	0.291(0.648)	0.250(0.448)	0.180(0.281)
	3	0.026(0.996)	0.114(0.956)	0.206(0.853)	0.250(0.698)	0.240(0.521)
	4	0.004(1.000)	0.035(0.991)	0.100(0.953)	0.172(0.870)	0.220(0.742)
	5		0.008(0.999)	0.035(0.988)	0.086(0.956)	0.147(0.888)
	6		0.001(1.000)	0.009(0.998)	0.032(0.988)	0.073(0.962)
	7			0.002(1.000)	0.009(0.998)	0.028(0.990)
	8				0.002(1.000)	0.008(0.998)
	9					0.002(1.000)
	10
	11
	12
	13
	14
15	0	0.463(0.463)	0.206(0.206)	0.087(0.087)	0.035(0.035)	0.013(0.013)
	1	0.366(0.829)	0.343(0.549)	0.231(0.319)	0.132(0.167)	0.067(0.080)
	2	0.135(0.964)	0.267(0.816)	0.286(0.604)	0.231(0.398)	0.156(0.236)
	3	0.031(0.995)	0.129(0.944)	0.218(0.823)	0.250(0.648)	0.225(0.461)
	4	0.005(0.999)	0.043(0.987)	0.116(0.938)	0.188(0.836)	0.225(0.686)
	5	0.001(1.000)	0.010(0.998)	0.045(0.983)	0.103(0.939)	0.165(0.852)
	6		0.002(1.000)	0.013(0.996)	0.043(0.982)	0.092(0.943)
	7			0.003(0.999)	0.014(0.996)	0.039(0.983)
	8			0.001(1.000)	0.003(0.999)	0.013(0.996)
	9				0.001(1.000)	0.003(0.999)
	10					0.001(1.000)
	11
	12
	13
	14
	15

n	x	0.30	0.35	0.40	0.45	0.50
1	0	0.700(0.700)	0.650(0.650)	0.600(0.600)	0.550(0.550)	0.500(0.500)
	1	0.300(1.000)	0.350(1.000)	0.400(1.000)	0.450(1.000)	0.500(1.000)
2	0	0.490(0.490)	0.423(0.423)	0.360(0.360)	0.303(0.303)	0.250(0.250)
	1	0.420(0.910)	0.455(0.877)	0.480(0.840)	0.495(0.798)	0.500(0.750)
	2	0.090(1.000)	0.122(1.000)	0.160(1.000)	0.203(1.000)	0.250(1.000)
3	0	0.343(0.343)	0.275(0.275)	0.216(0.216)	0.166(0.166)	0.125(0.125)
	1	0.441(0.784)	0.444(0.718)	0.432(0.648)	0.408(0.575)	0.375(0.500)
	2	0.189(0.973)	0.239(0.957)	0.288(0.936)	0.334(0.909)	0.375(0.875)
	3	0.027(1.000)	0.043(1.000)	0.064(1.000)	0.091(1.000)	0.125(1.000)
4	0	0.240(0.240)	0.179(0.179)	0.130(0.130)	0.092(0.092)	0.062(0.062)
	1	0.412(0.652)	0.384(0.563)	0.346(0.475)	0.299(0.391)	0.250(0.313)
	2	0.265(0.916)	0.311(0.874)	0.346(0.821)	0.368(0.759)	0.375(0.688)
	3	0.076(0.992)	0.111(0.985)	0.154(0.974)	0.200(0.959)	0.250(0.938)
	4	0.008(1.000)	0.015(1.000)	0.026(1.000)	0.041(1.000)	0.062(1.000)
5	0	0.168(0.168)	0.116(0.116)	0.078(0.078)	0.050(0.050)	0.031(0.031)
	1	0.360(0.528)	0.312(0.428)	0.259(0.337)	0.206(0.256)	0.156(0.187)
	2	0.309(0.837)	0.336(0.765)	0.346(0.683)	0.337(0.593)	0.312(0.500)
	3	0.132(0.969)	0.181(0.946)	0.230(0.913)	0.276(0.869)	0.312(0.812)
	4	0.028(0.998)	0.049(0.995)	0.077(0.990)	0.113(0.982)	0.156(0.969)
	5	0.002(1.000)	0.005(1.000)	0.010(1.000)	0.018(1.000)	0.031(1.000)
6	0	0.118(0.118)	0.075(0.075)	0.047(0.047)	0.028(0.028)	0.016(0.016)
	1	0.303(0.420)	0.244(0.319)	0.187(0.233)	0.136(0.164)	0.094(0.109)
	2	0.324(0.744)	0.328(0.647)	0.311(0.544)	0.278(0.442)	0.234(0.344)
	3	0.185(0.930)	0.235(0.883)	0.276(0.821)	0.303(0.745)	0.313(0.656)
	4	0.060(0.989)	0.095(0.978)	0.138(0.959)	0.186(0.931)	0.234(0.891)
	5	0.010(0.999)	0.020(0.998)	0.037(0.996)	0.061(0.992)	0.094(0.984)
	6	0.001(1.000)	0.002(1.000)	0.004(1.000)	0.008(1.000)	0.016(1.000)
7	0	0.082(0.082)	0.049(0.049)	0.028(0.028)	0.015(0.015)	0.008(0.008)
	1	0.247(0.329)	0.185(0.234)	0.131(0.159)	0.087(0.102)	0.055(0.063)
	2	0.318(0.647)	0.298(0.532)	0.261(0.420)	0.214(0.316)	0.164(0.227)
	3	0.227(0.874)	0.268(0.800)	0.290(0.710)	0.292(0.608)	0.273(0.500)
	4	0.097(0.971)	0.144(0.944)	0.194(0.904)	0.239(0.847)	0.273(0.773)
	5	0.025(0.996)	0.047(0.991)	0.077(0.981)	0.117(0.964)	0.164(0.938)
	6	0.004(1.000)	0.008(0.999)	0.017(0.998)	0.032(0.996)	0.055(0.992)
	7		0.001(1.000)	0.002(1.000)	0.004(1.000)	0.008(1.000)
8	0	0.058(0.058)	0.032(0.032)	0.017(0.017)	0.008(0.008)	0.004(0.004)
	1	0.198(0.255)	0.137(0.169)	0.090(0.106)	0.055(0.063)	0.031(0.035)
	2	0.296(0.552)	0.259(0.428)	0.209(0.315)	0.157(0.220)	0.109(0.145)
	3	0.254(0.806)	0.279(0.706)	0.279(0.594)	0.257(0.477)	0.219(0.363)
	4	0.136(0.942)	0.188(0.894)	0.232(0.826)	0.263(0.740)	0.273(0.637)
	5	0.047(0.989)	0.081(0.975)	0.124(0.950)	0.172(0.912)	0.219(0.855)
	6	0.010(0.999)	0.022(0.996)	0.041(0.991)	0.070(0.982)	0.109(0.965)
	7	0.001(1.000)	0.003(1.000)	0.008(0.999)	0.016(0.998)	0.031(0.996)
	8			0.001(1.000)	0.002(1.000)	0.004(1.000)
9	0	0.040(0.040)	0.021(0.021)	0.010(0.010)	0.005(0.005)	0.002(0.002)
	1	0.156(0.196)	0.100(0.121)	0.060(0.071)	0.034(0.039)	0.018(0.020)
	2	0.267(0.463)	0.216(0.337)	0.161(0.232)	0.111(0.150)	0.070(0.090)
	3	0.267(0.730)	0.272(0.609)	0.251(0.483)	0.212(0.361)	0.164(0.254)
	4	0.172(0.901)	0.219(0.828)	0.251(0.733)	0.260(0.621)	0.246(0.500)
	5	0.074(0.975)	0.118(0.946)	0.167(0.901)	0.213(0.834)	0.246(0.746)
	6	0.021(0.996)	0.042(0.989)	0.074(0.975)	0.116(0.950)	0.164(0.910)
	7	0.004(1.000)	0.010(0.999)	0.021(0.996)	0.041(0.991)	0.070(0.980)
	8		0.001(1.000)	0.004(1.000)	0.008(0.999)	0.018(0.998)
	9				0.001(1.000)	0.002(1.000)
10	0	0.028(0.028)	0.013(0.013)	0.006(0.006)	0.003(0.003)	0.001(0.001)
	1	0.121(0.149)	0.072(0.086)	0.040(0.046)	0.021(0.023)	0.010(0.011)
	2	0.233(0.383)	0.176(0.262)	0.121(0.167)	0.076(0.100)	0.044(0.055)
	3	0.267(0.650)	0.252(0.514)	0.215(0.382)	0.166(0.266)	0.117(0.172)
	4	0.200(0.850)	0.238(0.751)	0.251(0.633)	0.238(0.504)	0.205(0.377)
	5	0.103(0.953)	0.154(0.905)	0.201(0.834)	0.234(0.738)	0.246(0.623)
	6	0.037(0.989)	0.069(0.974)	0.111(0.945)	0.160(0.898)	0.205(0.828)
	7	0.009(0.998)	0.021(0.995)	0.042(0.988)	0.075(0.973)	0.117(0.945)
	8	0.001(1.000)	0.004(0.999)	0.011(0.998)	0.023(0.995)	0.044(0.989)
	9		0.001(1.000)	0.002(1.000)	0.004(1.000)	0.010(0.999)
	10					0.001(1.000)
11	0	0.020(0.020)	0.009(0.009)	0.004(0.004)	0.001(0.001)
	1	0.093(0.113)	0.052(0.061)	0.027(0.030)	0.013(0.014)	0.005(0.006)
	2	0.200(0.313)	0.140(0.200)	0.089(0.119)	0.051(0.065)	0.027(0.033)
	3	0.257(0.570)	0.225(0.426)	0.177(0.296)	0.126(0.191)	0.081(0.113)
	4	0.220(0.790)	0.243(0.668)	0.236(0.533)	0.206(0.397)	0.161(0.274)
	5	0.132(0.922)	0.183(0.851)	0.221(0.753)	0.236(0.633)	0.226(0.500)
	6	0.057(0.978)	0.099(0.950)	0.147(0.901)	0.193(0.826)	0.226(0.726)
	7	0.017(0.996)	0.038(0.988)	0.070(0.971)	0.113(0.939)	0.161(0.887)
	8	0.004(0.999)	0.010(0.998)	0.023(0.994)	0.046(0.985)	0.081(0.967)
	9	0.001(1.000)	0.002(1.000)	0.005(0.999)	0.013(0.998)	0.027(0.994)
	10			0.001(1.000)	0.002(1.000)	0.005(1.000)
	11
12	0	0.014(0.014)	0.006(0.006)	0.002(0.002)	0.001(0.001)
	1	0.071(0.085)	0.037(0.042)	0.017(0.020)	0.008(0.008)	0.003(0.003)
	2	0.168(0.253)	0.109(0.151)	0.064(0.083)	0.034(0.042)	0.016(0.019)
	3	0.240(0.493)	0.195(0.347)	0.142(0.225)	0.092(0.134)	0.054(0.073)
	4	0.231(0.724)	0.237(0.583)	0.213(0.438)	0.170(0.304)	0.121(0.194)
	5	0.158(0.882)	0.204(0.787)	0.227(0.665)	0.222(0.527)	0.193(0.387)
	6	0.079(0.961)	0.128(0.915)	0.177(0.842)	0.212(0.739)	0.226(0.613)
	7	0.029(0.991)	0.059(0.974)	0.101(0.943)	0.149(0.888)	0.193(0.806)
	8	0.008(0.998)	0.020(0.994)	0.042(0.985)	0.076(0.964)	0.121(0.927)
	9	0.001(1.000)	0.005(0.999)	0.012(0.997)	0.028(0.992)	0.054(0.981)
	10		0.001(1.000)	0.002(1.000)	0.007(0.999)	0.016(0.997)
	11				0.001(1.000)	0.003(1.000)
	12
13	0	0.010(0.010)	0.004(0.004)	0.001(0.001)
	1	0.054(0.064)	0.026(0.030)	0.011(0.013)	0.004(0.005)	0.002(0.002)
	2	0.139(0.202)	0.084(0.113)	0.045(0.058)	0.022(0.027)	0.010(0.011)
	3	0.218(0.421)	0.165(0.278)	0.111(0.169)	0.066(0.093)	0.035(0.046)
	4	0.234(0.654)	0.222(0.501)	0.184(0.353)	0.135(0.228)	0.087(0.133)
	5	0.180(0.835)	0.215(0.716)	0.221(0.574)	0.199(0.427)	0.157(0.291)
	6	0.103(0.938)	0.155(0.871)	0.197(0.771)	0.217(0.644)	0.209(0.500)
	7	0.044(0.982)	0.083(0.954)	0.131(0.902)	0.177(0.821)	0.209(0.709)
	8	0.014(0.996)	0.034(0.987)	0.066(0.968)	0.109(0.930)	0.157(0.867)
	9	0.003(0.999)	0.010(0.997)	0.024(0.992)	0.050(0.980)	0.087(0.954)
	10	0.001(1.000)	0.002(1.000)	0.006(0.999)	0.016(0.996)	0.035(0.989)
	11			0.001(1.000)	0.004(0.999)	0.010(0.998)
	12					0.002(1.000)
	13
14	0	0.007(0.007)	0.002(0.002)	0.001(0.001)
	1	0.041(0.047)	0.018(0.021)	0.007(0.008)	0.003(0.003)	0.001(0.001)
	2	0.113(0.161)	0.063(0.084)	0.032(0.040)	0.014(0.017)	0.006(0.006)
	3	0.194(0.355)	0.137(0.220)	0.085(0.124)	0.046(0.063)	0.022(0.029)
	4	0.229(0.584)	0.202(0.423)	0.155(0.279)	0.104(0.167)	0.061(0.090)
	5	0.196(0.781)	0.218(0.641)	0.207(0.486)	0.170(0.337)	0.122(0.212)
	6	0.126(0.907)	0.176(0.816)	0.207(0.692)	0.209(0.546)	0.183(0.395)
	7	0.062(0.969)	0.108(0.925)	0.157(0.850)	0.195(0.741)	0.209(0.605)
	8	0.023(0.992)	0.051(0.976)	0.092(0.942)	0.140(0.881)	0.183(0.788)
	9	0.007(0.998)	0.018(0.994)	0.041(0.982)	0.076(0.957)	0.122(0.910)
	10	0.001(1.000)	0.005(0.999)	0.014(0.996)	0.031(0.989)	0.061(0.971)
	11		0.001(1.000)	0.003(0.999)	0.009(0.998)	0.022(0.994)
	12			0.001(1.000)	0.002(1.000)	0.006(0.999)
	13					0.001(1.000)
	14
15	0	0.005(0.005)	0.002(0.002)
	1	0.031(0.035)	0.013(0.014)	0.005(0.005)	0.002(0.002)
	2	0.092(0.127)	0.048(0.062)	0.022(0.027)	0.009(0.011)	0.003(0.004)
	3	0.170(0.297)	0.111(0.173)	0.063(0.091)	0.032(0.042)	0.014(0.018)
	4	0.219(0.515)	0.179(0.352)	0.127(0.217)	0.078(0.120)	0.042(0.059)
	5	0.206(0.722)	0.212(0.564)	0.186(0.403)	0.140(0.261)	0.092(0.151)
	6	0.147(0.869)	0.191(0.755)	0.207(0.610)	0.191(0.452)	0.153(0.304)
	7	0.081(0.950)	0.132(0.887)	0.177(0.787)	0.201(0.654)	0.196(0.500)
	8	0.035(0.985)	0.071(0.958)	0.118(0.905)	0.165(0.818)	0.196(0.696)
	9	0.012(0.996)	0.030(0.988)	0.061(0.966)	0.105(0.923)	0.153(0.849)
	10	0.003(0.999)	0.010(0.997)	0.024(0.991)	0.051 (0.975)	0.092(0.941)
	11	0.001(1.000)	0.002(1.000)	0.007(0.998)	0.019(0.994)	0.042(0.982)
	12			0.002(1.000)	0.005(0.999)	0.014(0.996)
	13				0.001(1.000)	0.003(1.000)
	14
	15

n	x	0.55	0.60	0.65	0.70	0.75
1	0	0.450(0.450)	0.400(0.400)	0.350(0.350)	0.300(0.300)	0.250(0.250)
	1	0.550(1.000)	0.600(1.000)	0.650(1.000)	0.700(1.000)	0.750(1.000)
	20	0.202(0.202)	0.160(0.160)	0.122(0.122)	0.090(0.090)	0.062(0.062)
	1	0.495(0.698)	0.480(0.640)	0.455(0.577)	0.420(0.510)	0.375(0.438)
	2	0.303(1.000)	0.360(1.000)	0.423(1.000)	0.490(1.000)	0.562(1.000)
3	0	0.091(0.091)	0.064(0.064)	0.043(0.043)	0.027(0.027)	0.016(0.016)
	1	0.334(0.425)	0.288(0.352)	0.239(0.282)	0.189(0.216)	0.141(0.156)
	2	0.408(0.834)	0.432(0.784)	0.444(0.725)	0.441(0.657)	0.422(0.578)
	3	0.166(1.000)	0.216(1.000)	0.275(1.000)	0.343(1.000)	0.422(1.000)
4	0	0.041(0.041)	0.026(0.026)	0.015(0.015)	0.008(0.008)	0.004(0.004)
	1	0.200(0.241)	0.154(0.179)	0.111(0.126)	0.076(0.084)	0.047(0.051)
	2	0.368(0.609)	0.346(0.525)	0.311(0.437)	0.265(0.348)	0.211(0.262)
	3	0.299(0.908)	0.346(0.870)	0.384(0.821)	0.412(0.760)	0.422(0.684)
	4	0.092(1.000)	0.130(1.000)	0.179(1.000)	0.240(1.000)	0.316(1.000)
5	0	0.018(0.018)	0.010(0.010)	0.005(0.005)	0.002(0.002)	0.001(0.001)
	1	0.113(0.131)	0.077(0.087)	0.049(0.054)	0.028(0.031)	0.015(0.016)
	2	0.276(0.407)	0.230(0.317)	0.181(0.235)	0.132(0.163)	0.088(0.104)
	3	0.337(0.744)	0.346(0.663)	0.336(0.572)	0.309(0.472)	0.264(0.367)
	4	0.206(0.950)	0.259(0.922)	0.312(0.884)	0.360(0.832)	0.396(0.763)
	5	0.050(1.000)	0.078(1.000)	0.116(1.000)	0.168(1.000)	0.237(1.000)
6	0	0.008(0.008)	0.004(0.004)	0.002(0.002)	0.001(0.001)
	1	0.061(0.069)	0.037(0.041)	0.020(0.022)	0.010(0.011)	0.004(0.005)
	2	0.186(0.255)	0.138(0.179)	0.095(0.117)	0.060(0.070)	0.033(0.038)
	3	0.303(0.558)	0.276(0.456)	0.235(0.353)	0.185(0.256)	0.132(0.169)
	4	0.278(0.836)	0.311(0.767)	0.328(0.681)	0.324(0.580)	0.297(0.466)
	5	0.136(0.972)	0.187(0.953)	0.244(0.925)	0.303(0.882)	0.356(0.822)
	6	0.028(1.000)	0.047(1.000)	0.075(1.000)	0.118(1.000)	0.178(1.000)
7	0	0.004(0.004)	0.002(0.002)	0.001(0.001)
	1	0.032(0.036)	0.017(0.019)	0.008(0.009)	0.004(0.004)	0.001(0.001)
	2	0.117(0.153)	0.077(0.096)	0.047(0.056)	0.025(0.029)	0.012(0.013)
	3	0.239(0.392)	0.194(0.290)	0.144(0.200)	0.097(0.126)	0.058(0.071)
	4	0.292(0.684)	0.290(0.580)	0.268(0.468)	0.227(0.353)	0.173(0.244)
	5	0.214(0.898)	0.261(0.841)	0.298(0.766)	0.318(0.671)	0.311(0.555)
	6	0.087(0.985)	0.131(0.972)	0.185(0.951)	0.247(0.918)	0.311(0.867)
	7	0.015(1.000)	0.028(1.000)	0.049(1.000)	0.082(1.000)	0.133(1.000)
8	0	0.002(0.002)	0.001(0.001)
	1	0.016(0.018)	0.008(0.009)	0.003(0.004)	0.001(0.001)
	2	0.070(0.088)	0.041(0.050)	0.022(0.025)	0.010(0.011)	0.004(0.004)
	3	0.172(0.260)	0.124(0.174)	0.081(0.106)	0.047(0.058)	0.023(0.027)
	4	0.263(0.523)	0.232(0.406)	0.188(0.294)	0.136(0.194)	0.087(0.114)
	5	0.257(0.780)	0.279(0.685)	0.279(0.572)	0.254(0.448)	0.208(0.321)
	6	0.157(0.937)	0.209(0.894)	0.259(0.831)	0.296(0.745)	0.311(0.633)
	7	0.055(0.992)	0.090(0.983)	0.137(0.968)	0.198(0.942)	0.267(0.900)
	8	0.008(1.000)	0.017(1.000)	0.032(1.000)	0.058(1.000)	0.100(1.000)
9	0	0.001(0.001)
	1	0.008(0.009)	0.004(0.004)	0.001(0.001)
	2	0.041(0.050)	0.021(0.025)	0.010(0.011)	0.004(0.004)	0.001(0.001)
	3	0.116(0.166)	0.074(0.099)	0.042(0.054)	0.021(0.025)	0.009(0.010)
	4	0.213(0.379)	0.167(0.267)	0.118(0.172)	0,074(0.099)	0.039(0.049)
	5	0.260(0.639)	0.251(0.517)	0.219(0.391)	0.172(0.270)	0.117(0.166)
	6	0.212(0,850)	0.251(0.768)	0,272(0,663)	0.267(0.537)	0.234(0.399)
	7	0.111(0.961)	0.161(0.929)	0,216(0,879)	0.267(0.804)	0.300(0.700)
	8	0.034(0.995)	0.060(0.990)	0.100(0.979)	0.156(0.960)	0.225(0.925)
	9	0.005(1.000)	0.010(1.000)	0.021(1.000)	0.040(1.000)	0.075(1.000)
10	0
	1	0.004(0.005)	0.002(0.002)	0.001(0.001)
	2	0.023(0.027)	0.011(0.012)	0.004(0,005)	0.001(0.002)
	3	0.075(0.102)	0.042(0.055)	0.021(0.026)	0.009(0.011)	0.003(0.004)
	4	0.160(0.262)	0.111(0.166)	0.069(0.095)	0.037(0.047)	0.016(0.020)
	5	0.234(0.496)	0.201(0.367)	0.154(0.249)	0.103(0.150)	0.058(0.078)
	6	0.238(0.734)	0,251(0,618)	0.238(0,486)	0.200(0.350)	0,146(0.224)
	7	0.166(0.900)	0.215(0.833)	0.252(0,738)	0.267(0.617)	0,250(0.474)
	8	0.076(0.977)	0.121(0.954)	0.176(0.914)	0.233(0.851)	0.282(0.756)
	9	0.021(0.997)	0.040(0.994)	0.072(0.987)	0.121(0.972)	0.188(0.944)
	10	0.003(1.000)	0.006(1.000)	0.013(1.000)	0.028(1.000)	0.056(1.000)
11	0
	1	0.002(0.002)	0,001(0.001)
	2	0.013(0.015)	0.005(0.006)	0.002(0.002)	0.001(0.001)
	3	0.046(0.061)	0.023(0.029)	0.010(0.012)	0.004(0.004)	0.001(0.001)
	4	0.113(0.174)	0.070(0.099)	0.038(0.050)	0.017(0.022)	0.006(0.008)
	5	0.193(0.367)	0.147(0.247)	0.099(0.149)	0.057(0.078)	0.027(0.034)
	6	0.236(0.603)	0.221(0.467)	0.183(0.332)	0.132(0.210)	0.080(0.115)
	7	0.206(0.809)	0.236(0.704)	0.243(0.574)	0,220(0.430)	0.172(0.287)
	8	0.126(0.935)	0.177(0,881)	0.225(0.800)	0,257(0,687)	0.258(0.545)
	9	0.051(0.986)	0.089(0.970)	0.140(0.939)	0,200(0,887)	0.258(0.803)
	10	0.013(0.999)	0.027(0.996)	0.052(0,991)	0.093(0,980)	0.155(0,958)
	11	0.001(1.000)	0.004(1.000)	0.009(1.000)	0.020(1.000)	0.042(1.000)
12	0
	1	0.001(0.001)
	2	0.007(0.008)	0.002(0.003)	0.001(0.001)
	3	0.028(0.036)	0.012(0.015)	0.005(0.006)	0.001(0.002)
	4	0.076(0.112)	0.042(0.057)	0.020(0.026)	0.008(0.009)	0,002(0,003)
	5	0.149(0.261)	0.101(0.158)	0.059(0.085)	0.029(0.039)	0.011(0.014)
	6	0.212(0.473)	0.177(0.335)	0.128(0.213)	0.079(0.118)	0.040(0.054)
	7	0.222(0.696)	0.227(0.562)	0.204(0.417)	0.158(0.276)	0.103(0.158)
	8	0.170(0.866)	0.213(0.775)	0.237(0.653)	0.231(0.507)	0.194(0.351)
	9	0.092(0.958)	0.142(0.917)	0.195(0.849)	0.240(0.747)	0.258(0,609)
	10	0.034(0.992)	0.064(0.980)	0.109(0.958)	0.168(0.915)	0,232(0.842)
	11	0.008(0.999)	0.017(0.998)	0.037(0.994)	0.071(0.986)	0.127(0.968)
	12	0.001(1.000)	0.002(1.000)	0.006(1.000)	0.014(1.000)	0.032(1.000)
13	0
	1
	2	0.004(0.004)	0.001(0.001)
	3	0.016(0.020)	0.006(0.008)	0.002(0.003)	0.001(0.001)
	4	0.050(0.070)	0.024(0.032)	0.010(0.013)	0.003(0.004)	0.001(0.001)
	5	0.109(0.179)	0.066(0.098)	0.034(0.046)	0.014(0.018)	0.005(0.006)
	6	0.177(0.356)	0.131(0.229)	0.083(0.129)	0.044(0.062)	0.019(0.024)
	7	0.217(0.573)	0.197(0.426)	0.155(0.284)	0.103(0.165)	0.056(0.080)
	8	0.199(0.772)	0.221(0.647)	0.215(0.499)	0.180(0.346)	0.126(0.206)
	9	0.135(0.907)	0.184(0.831)	0.222(0.722)	0.234(0.579)	0.210(0.416)
	10	0.066(0.973)	0.111(0.942)	0.165(0.887)	0.218(0.798)	0.252(0.667)
	11	0.022(0.995)	0.045(0.987)	0.084(0.970)	0.139(0.936)	0.206(0.873)
	12	0.004(1.000)	0.011(0.999)	0.026(0.996)	0.054(0.990)	0.103(0.976)
	13		0.001(1.000)	0.004(1.000)	0.010(1.000)	0.024(1.000)
14	0
	1
	2	0.002(0.002)	0.001(0.001)
	3	0.009(0.011)	0.003(0.004)	0.001(0.001)
	4	0.031(0.043)	0.014(0.018)	0.005(0.006)	0.001(0.002)
	5	0.076(0.119)	0.041(0.058)	0.018(0.024)	0.007(0.008)	0.002(0.002)
	6	0.140(0.259)	0.092(0.150)	0.051(0.075)	0.023(0.031)	0.008(0.010)
	7	0.195(0.454)	0.157(0.308)	0.108(0.184)	0.062(0.093)	0.028(0.038)
	8	0.209(0.663)	0.207(0.514)	0.176(0.359)	0.126(0.219)	0.073(0.112)
	9	0.170(0.833)	0.207(0.721)	0.218(0.577)	0.196(0.416)	0.147(0.258)
	10	0.104(0.937)	0.155(0.876)	0.202(0.780)	0.229(0.645)	0.220(0.479)
	11	0.046(0.983)	0.085(0.960)	0.137(0.916)	0.194(0.839)	0.240(0.719)
	12	0.014(0.997)	0.032(0.992)	0.063(0.979)	0.113(0.953)	0.180(0.899)
	13	0.003(1.000)	0.007(0.999)	0.018(0.998)	0.041(0.993)	0.083(0.982)
	14		0.001(1.000)	0.002(1.000)	0.007(1.000)	0.018(1.000)
15	0
	1
	2	0.001(0.001)
	3	0.005(0.006)	0.002(0.002)
	4	0.019(0.025)	0.007(0.009)	0.002(0.003)	0.001(0.001)
	5	0.051(0.077)	0.024(0.034)	0.010(0.012)	0.003(0.004)	0.001(0.001)
	6	0.105(0.182)	0.061(0.095)	0.030(0.042)	0.012(0.015)	0.003(0.004)
	7	0.165(0.346)	0.118(0.213)	0.071(0.113)	0.035(0.050)	0.013(0.017)
	8	0.201(0.548)	0.177(0.390)	0.132(0.245)	0.081(0.131)	0.039(0.057)
	9	0.191(0.739)	0.207(0.597)	0.191(0.436)	0.147(0.278)	0.092(0.148)
	10	0.140(0.880)	0.186(0.783)	0.212(0.648)	0.206(0.485)	0.165(0.314)
	11	0.078(0.958)	0.127(0.909)	0.179(0.827)	0.219(0.703)	0.225(0.539)
	12	0.032(0.989)	0.063(0.973)	0.111(0.938)	0.170(0.873)	0.225(0.764)
	13	0.009(0.998)	0.022(0.995)	0.048(0.986)	0.092(0.965)	0.156(0.920)
	14	0.002(1.000)	0.005(1.000)	0.013(0.998)	0.031(0.995)	0.067(0.987)
	15			0.002(1.000)	0.005(1.000)	0.013(1.000)

n	x	0.80	0.85	0.90	0.95
1	0	0.200(0.200)	0.150(0.150)	0.100(0.100)	0.050(0.050)
	1	0.800(1.000)	0.850(1.000)	0.900(1.000)	0.950(1.000)
2	0	0.040(0.040)	0.023(0.023)	0.010(0.010)	0.003(0.003)
	1	0.320(0.360)	0.255(0.278)	0.180(0.190)	0.095(0.098)
	2	0.640(1.000)	0.722(1.000)	0.810(1.000)	0.902(1.000)
3	0	0.008(0.008)	0.003(0.003)	0.001(0.001)
	1	0.096(0.104)	0.057(0.061)	0.027(0.028)	0.007(0.007)
	2	0.384(0.488)	0.325(0.386)	0.243(0.271)	0.135(0.143)
	3	0.512(1.000)	0.614(1.000)	0.729(1.000)	0.857(1.000)
4	0	0.002(0.002)	0.001(0.001)
	1	0.026(0.027)	0.011(0.012)	0.004(0.004)
	2	0.154(0.181)	0.098(0.110)	0.049(0.052)	0,014(0.014)
	3	0.410(0.590)	0.368(0.478)	0.292(0.344)	0.171(0.185)
	4	0.410(1.000)	0.522(1.000)	0.656(1.000)	0.815(1.000)
5	0
	1	0.006(0.007)	0.002(0.002)
	2	0.051(0.058)	0.024(0.027)	0.008(0.009)	0.001(0.001)
	3	0.205(0.263)	0.138(0.165)	0.073(0.081)	0.021(0.023)
	4	0.410(0.672)	0.392(0.556)	0.328(0.410)	0.204(0.226)
	5	0.328(1.000)	0.444(1.000)	0.590(1.000)	0.774(1.000)
6	0
	1	0.002(0.002)
	2	0.015(0.017)	0.005(0.006)	0.001(0.001)
	3	0.082(0.099)	0.041(0.047)	0.015(0.016)	0.002(0.002)
	4	0.246(0.345)	0.176(0.224)	0.098(0.114)	0.031(0.033)
	5	0.393(0.738)	0.399(0.623)	0.354(0.469)	0.232(0.265)
	6	0.262(1.000)	0.377(1.000)	0.531(1.000)	0.735(1.000)
7	0
	1
	2	0.004(0.005)	0.001(0.001)
	3	0.029(0.033)	0.011(0.012)	0.003(0.003)
	4	0.115(0.148)	0.062(0.074)	0.023(0.026)	0.004(0.004)
	5	0.275(0.423)	0.210(0.283)	0.124(0.150)	0.041(0.044)
	6	0.367(0.790)	0.396(0.679)	0.372(0.522)	0.257(0.302)
	7	0.210(1.000)	0.321(1.000)	0.478(1.000)	0.698(1.000)
8	0
	1
	2	0.001(0.001)
	3	0.009(0.010)	0.003(0.003)
	4	0.046(0.056)	0.018(0.021)	0.005(0.005)
	5	0.147(0.203)	0.084(0.105)	0.033(0.038)	0.005(0.006)
	6	0.294(0.497)	0.238(0.343)	0.149(0.187)	0.051(0.057)
	7	0.336(0.832)	0.385(0.728)	0.383(0.570)	0.279(0.337)
	8	0.168(1.000)	0.272(1.000)	0.430(1.000)	0.663(1.000)
9	0
	1
	2
	3	0.003(0.003)	0.001(0.001)
	4	0.017(0.020)	0.005(0.006)	0.001(0.001)
	5	0.066(0.086)	0.028(0.034)	0.007(0.008)	0.001(0.001)
	6	0.176(0.262)	0.107(0.141)	0.045(0.053)	0.008(0.008)
	7	0.302(0.564)	0.260(0.401)	0.172(0.225)	0.063(0.071)
	8	0.302(0.866)	0.368(0.768)	0.387(0.613)	0.299(0.370)
	9	0.134(1.000)	0.232(1.000)	0.387(1.000)	0.630(1.000)
10	0
	1
	2
	3	0.001(0.001)
	4	0.006(0.006)	0.001(0.001)
	5	0.026(0.033)	0.008(0.010)	0.001(0.002)
	6	0.088(0.121)	0.040(0.050)	0.011(0.013)	0.001(0.001)
	7	0.201(0.322)	0.130(0.180)	0.057(0.070)	0.010(0.012)
	8	0.302(0.624)	0.276(0.456)	0.194(0.264)	0.075(0.086)
	9	0.268(0.893)	0.347(0.803)	0.387(0.651)	0.315(0.401)
	10	0.107(1.000)	0.197(1.000)	0.349(1.000)	0.599(1.000)
11	0
	1
	2
	3
	4	0.002(0.002)
	5	0.010(0.012)	0.002(0.003)
	6	0.039(0.050)	0.013(0.016)	0.002(0.003)
	7	0.111(0.161)	0.054(0.069)	0.016(0.019)	0.001(0.002)
	8	0.221(0.383)	0.152(0.221)	0.071(0.090)	0.014(0.015)
	9	0.295(0.678)	0.287(0.508)	0.213(0.303)	0.087(0.102)
	10	0.236(0.914)	0.325(0.833)	0.384(0.686)	0.329(0.431)
	11	0.086(1.000)	0.167(1.000)	0.314(1.000)	0.569(1.000)
12	0
	1
	2
	3
	4	0.001(0.001)
	5	0.003(0.004)	0.001(0.001)
	6	0.016(0.019)	0.004(0.005)
	7	0.053(0.073)	0.019(0.024)	0.004(0.004)
	8	0.133(0.205)	0.068(0.092)	0.021(0.026)	0.002(0.002)
	9	0.236(0.442)	0.172(0.264)	0.085(0.111)	0.017(0.020)
	10	0.283(0.725)	0.292(0.557)	0.230(0.341)	0.099(0.118)
	11	0.206(0.931)	0.301(0.858)	0.377(0.718)	0.341(0.460)
	12	0.069(1.000)	0.142(1.000)	0.282(1.000)	0.540(1.000)
13	0
	1
	2
	3
	4
	5	0.001(0.001)
	6	0.006(0.007)	0.001(0.001)
	7	0.023(0.030)	0.006(0.008)	0.001(0.001)
	8	0.069(0.099)	0.027(0.034)	0.006(0.006)
	9	0.154(0.253)	0.084(0.118)	0.028(0.034)	0.003(0.003)
	10	0.246(0.498)	0.190(0.308)	0.100(0.134)	0.021(0.025)
	11	0.268(0.766)	0.294(0.602)	0.245(0.379)	0.111(0.135)
	12	0.179(0.945)	0.277(0.879)	0.367(0.746)	0.351(0.487)
	13	0.055(1.000)	0.121(1.000)	0.254(1.000)	0.513(1.000)
14	0
	1
	2
	3
	4
	5
	6	0.002(0.002)
	7	0.009(0.012)	0.002(0.002)
	8	0.032(0.044)	0.009(0.012)	0.001(0.001)
	9	0.086(0.130)	0.035(0.047)	0.008(0.009)
	10	0.172(0.302)	0.100(0.147)	0.035(0.044)	0.004(0.004)
	11	0.250(0.552)	0.206(0.352)	0.114(0.158)	0.026(0.030)
	12	0.250(0.802)	0.291(0.643)	0.257(0.415)	0.123(0.153)
	13	0.154(0.956)	0.254(0.897)	0.356(0.771)	0.359(0.512)
	14	0.044(1.000)	0.103(1.000)	0.229(1.000)	0.488(1.000)
15	0
	1
	2
	3
	4
	5
	6	0.001(0.001)
	7	0.003(0.004)	0.001(0.001)
	8	0.014(0.018)	0.003(0.004)
	9	0.043(0.061)	0.013(0.017)	0.002(0.002)
	10	0.103(0.164)	0.045(0.062)	0.010(0.013)	0.001(0.001)
	11	0.188(0.352)	0.116(0.177)	0.043(0.056)	0.005(0.005)
	12	0.250(0.602)	0.218(0.396)	0.129(0.184)	0.031(0.036)
	13	0.231(0.833)	0.286(0.681)	0.267(0.451)	0.135(0.171)
	14	0.132(0.965)	0.231(0.913)	0.343(0.794)	0.366(0.537)
	15	0.035(1.000)	0.087(1.000)	0.206(1.000)	0.463(1.000)