Hack 63. Sense the Real Randomness of Life

Before you accuse the casino of running a crooked game or threaten your boss with a lawsuit for hiring only blonde women, here's a tool for separating those nonrandom-seeming situations that probably did occur randomly from those nonrandom-seeming situations that probably did not occur randomly. Probably.

As you become more and more aware of the role that chance plays in the world around you, and begin to habitually stat-hack your way through everyday situations, you might become overly sensitive to patterns that don't seem right. Don't abuse your newfound powers, though, and treat probabilities as certainties. Additionally, don't make the mistake of expecting events that are supposed to be random to look random.

What Does Random Look Like?

Looking random and being random are not the same things. When events have several possible and equally likely outcomes, any of them can happen. The way the human mind works, though, many people think that the pattern of outcomes of events with several equally likely outcomes ought to look a certain way, a way that somehow looks random (whatever that means).

For example, real-world research has found that people tend to believe that, when flipping coins, the most probable outcomes are those that look the most mixed up. To illustrate this idea, look at Table 6-2. (Avoid looking at Table 6-3 until you have read a bit more.) Which exact sequence of coin flips do you think is most likely to occur?

Table Coin-flip patterns, with probabilities not shown

Answer	Pattern of heads ands tails	Probability
A	Heads, Tails, Heads, Heads, Tails	?
B	Tails, Tails, Tails, Tails, Tails	?
C	Heads, Heads, Tails, Tails, Tails	?
D	Heads, Heads, Heads, Heads, Tails	?

Many people give the answer "A." Maybe you did, too. When asked to explain why A seems the most likely outcome, the answers include statements like these:

• "The others are too ordered."

• "A is more mixed up, so it's more likely."

• "A looks more random, like it could really happen."

Even though you know that coin flipping is random (assuming the coin isn't weighted), looking random doesn't make something more probable. All of these patterns of coin flips are actually equally probable, as shown by the math in Table 6-3.

Table Coin-flip patterns, with probabilities

Answer	Pattern of heads and tails	Probability
A	Heads, Tails, Heads, Heads, Tails	1/2x1/2x1/2x1/2x1/2 = 1/32 = .03125
B	Tails, Tails, Tails, Tails, Tails	1/2x1/2x1/2x1/2x1/2 = 1/32 = .03125
C	Heads, Heads, Tails, Tails, Tails	1/2x1/2x1/2x1/2x1/2 = 1/32 = .03125
D	Heads, Heads, Heads, Heads, Tails	1/2x1/2x1/2x1/2x1/2 = 1/32 = .03125

When asked to predict a specific outcome of a series of coin flips, all possible outcomes must be equal, because each flip of the coin is independent of the other flips. In other words, the coin doesn't know whether it just landed on Heads or Tails, so there is no way that the coin can know which side it is supposed to land on the next time it is flipped. A coin, like dice or a roulette wheel, has no memory.

How to Spot Random Outcomes

To know an unusual sequence of events when you see it, you need to decide whether you are supposed to be paying attention to a combination or a permutation. In probability theory, we talk about calculating odds by looking at the probabilities of certain combinations (say, three Heads and two Tails in any order) and the probabilities of certain permutations (an exact sequence that would result in three Heads and two Tails, such as Heads, Tails, Heads, Heads, Tails, in that particular order).

If you are asked a question about which outcome is the most likely, or whether a given outcome could have occurred by chance, first determine whether you are being asked about combinations (the total number of Heads and Tails in any order, for example, or the number of different ways of drawing five playing cards of the same suit) or about the permutations that are possible. Here are the important distinctions between the two:

Combinations

A combination is the total number of ways that one could end up with a particular number of values when drawing randomly from some population. Coin flips are samples drawn from a theoretically infinitely large population made up of 50 percent Heads and 50 percent Tails. The number of combinations varies, depending on the number of a certain value one is interested in. In other words, with five draws or flips, there are more ways to draw out three heads than there are ways to draw out five heads. So, drawing three heads is likelier than five heads.

Permutations

Permutations are the number of ways that a given number of elements could be arranged. In other words, they are the number of exact sequences. In our coin-flip example, 5 elements that can each be 1 of 2 values results in 32 different possible orders of arrangement. So, each of the permutations shown in Table 6-3 will occur 1 out of every 32 times.

How to Calculate Combinations

The number of possible combinations is calculated by taking the number of possible values for one draw (e.g., two values for a coin: Heads or Tails) and multiplying it by itself for each draw:

There are 32 possible combinations of 5 coin flips (2⁵).

The equation for computing the number of ways to get a particular draw (e.g., three Heads) out of a particular number of elements drawn from a population is:

The previous equation requires these variables:

n

The number of elements or draws (e.g., 5 coin flips).

r

The particular draw of interest (e.g., 3 Heads).

!

Factorial, which means to take the number and multiply it by that number minus 1, then by that number minus 2, and so on, all the way down to 1. For example, 5! represents 5x4x3x2x1 = 120 (which, by the way, is why there are 120 possible combinations of five cards in a poker hand [Hack #62]).

So, the number of ways to get three Heads out of five coin flips is:

10 combinations out of 32 possible combinations means that you will get exactly 3 heads by flipping a coin 5 times 10/32 times, or about 31 percent of the time.

Statistics Hacking on a Desert Island

If you were on a desert island and didn't have access to books or equations and had to find out how often exactly three heads should come up in a group of five coin flips, you could use the brute force method of listing all the possible patterns of flips and counting how many of them have exactly three heads. It would look like this, with the outcome of interest (three heads) shown in bold:

HHHHH THHHH HHHHT THHHT HHTTH THTTH HHTTT THTTT HHHTH THHTH HHHTT THHTT HHTHH THTHH HHTHT THTHT HTHHH TTHHH HTHHT TTHHT HTTTH TTTTH HTHTT TTHTT HTTHH TTTHH HTTTT TTTTT HTHTH TTHTH HTTHT TTTHT

When to Be Suspicious

Deciding whether a pattern is random (i.e., what one would expect by chance) is a matter of:

• Knowing the chances of certain combinations (not permutations)

• Fighting the psychological tendency to expect chance results to not produce a recognizable pattern

• Setting a standard for how unlikely an event must be before questioning the data

Let's return to our table of coin flips, shown now in Table 6-4 with the added chances of certain outcomes of interest.

Table Coin-flip outcomes and probabilities

Order	Order probability	Outcome	Outcome probability
Heads, Tails, Heads, Heads, Tails	.03125	Three Heads	.31250
Tails, Tails, Tails, Tails, Tails	.03125	Five Tails	.03125
Heads, Heads, Tails, Tails, Tails	.03125	Three Tails	.31250
Heads, Heads, Heads, Heads, Tails	.03125	Four Heads	.15625

The rarest of these outcomes is five Tails, which will occur about 3 times for every 100 times you produce five coin flips. It is unlikely to happen by chance on a given attempt, but it will happen occasionally across a series of attempts. If it happens frequently across a series of attempts, something might be up.

What level of likelihood are you comfortable with? How rare must an event be before you decide it did not occur by chance? Scientists have set a standard of 5 percent. If study results suggest an outcome that would occur by chance only 5 percent or less of the time, it is usually considered to be significant, and is probably evidence that something other than chance is in play.

You get to decide for yourself, though, when you want to accuse someone of being a cheat. Good luck on making that decision! It should result in fist fights less than 5 percent of the time.

Jill Lohmeier with Bruce Frey