Coins and Independence

This section looks at the mathematics of flipping coins, and the next section looks at the mathematics of rolling dice. These examples can help to illustrate the more general rules of probability that will be discussed subsequently.

Coins

What is the probability of flipping a coin four times in a row and having it land heads each time? One way to solve this problem is to set up the sample space as the set of all possible sequences of coin flips. For example, one possible sequence is (H,T,H,T), where you get heads followed by tails followed by heads followed by tails. Overall, there are sixteen possible sequences:

(H,H,H,H), (H,H,H,T), (H,H,T,H), (H,H,T,T), (H,T,H,H), (H,T,H,T), (H,T,T,H), (H,T,T,T), (T,H,H,H), (T,H,H,T), (T,H,T,H), (T,H,T,T), (T,T,H,H), (T,T,H,T), (T,T,T,H), (T,T,T,T)

Of these sixteen sequences, only the first sequence has four heads. Since each sequence would seem to have an equal probability, the logical inference is that the probability of getting four straight heads is 1/16.

What is the probability of getting exactly three heads in four coin flips? Now, the calculation is not so simple. It turns out that of the sixteen sequences, four of them have three heads. The general answer to this question was given by Pascal.

Pascal's Triangle

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Each row of the triangle is constructed by adding the adjacent numbers in the preceding row, and then putting a one on the far left and far right.

For example, the second row is 1 2 1. When we add 1 and 2, we get 3, which we put in between them on the third row. Outside the two 3's, we extend the row by putting a 1 to the left and to the right.

Question: What will be the contents of the sixth row of the triangle ?

How else might we compute the probability of getting four heads in a row? One approach is to think in terms of a sequence of flips, where as soon as you get tails, you stop.

- The probability of coming up heads on the first flip is 1/2. If you get tails on the first flip, you might as well stop, because you cannot possibly get four heads. So, half the time you stop, and half the time you keep going.
- Assuming we kept going, then we flip the second coin. Again, the probability of heads is 1/2. Again, we only keep going if it comes up heads. So half the time we keep going. Overall, the chance that we will keep going is 1/2 of 1/2, or 1/4.
- By now, 3/4 of the time we will have stopped, and 1/4 of the time we will have moved on to flip a third coin. Again, the probability of heads is 1/2. So, the probability that we will keep going is 1/2 of 1/4, or 1/8.
- Finally, we have the fourth coin flip. We only get to this point 1/8 times. Again, the probability of heads is 1/2. The probability of four heads is thus 1/2 of 1/8, or 1/16.

As a shortcut, we could say that the probability of getting heads on any one throw is 1/2. The probability of getting four heads in a row therefore is (1/2)(1/2)(1/2(1/2), or (1/2)^{4}.

A general approach to analyzing coin flips is called Pascal's triangle (right). The triangle is a shortcut way to describe the sample space for the number of heads and tails from a sequence of coin tosses. The first row says that with one coin, we can have either all heads (1) or all tails (1).

The second row says that if we toss two coins, we have one chance of getting all heads, two chances of getting one heads and one tails, and one chance of getting all tails.

The third row says that if we toss three coins, we have one chance of getting all heads, three chances of getting one head and two tails, three chances of getting two heads and one tail, and one chance of getting three tails. Since the sum of the row is 8, the probability of getting two heads and one tail is 3/8.

Question: use the triangle to find the probability of getting exactly three heads in four coin tosses. What about exactly three heads in six coin tosses?

Implicitly, we are relying on the assumption that each coin flip is independent. When coin flips are independent, it means that the probability of a coin coming up heads does not depend in any way on previous coin flips.

Independence and Superstititious Sportscasters

Many people do not understand the concept of independence. Sportscasters are notorious for this.

For example, suppose that a sequence of seven coin flips came up with five heads and two tails. What is the probability of getting tails on the next coin flip? A sportscaster would say that "the law of averages" says that we are more likely to get tails.

But there is no law of averages! The chance of getting tails on the next coin flip is 1/2.

What is true is that we expect that as the number of coin flips gets large, the proportion of heads will become closer to 50 percent. However, that is because going forward, we expect about half of the flips to be tails, not because we expect the coin to know that it has an excess of three heads and it needs to come up tails more often to make up for it.

Here is another common superstition in sports. Suppose that Kobe the coin-flipper gets heads five times in a row. A sportscaster might say, "Kobe is really hot. The coach should give the coin to Kobe to make the next flip, because he probably can get heads again."

It isn't just sportscasters who are superstitious. The theories that people use to trade stocks have no statistical validity to them. What people do not realize (or cannot accept) is that the movement of a stock price tomorrow is independent of its movement today.

"Prices have no memory and yesterday has nothing to do with tomorrow. Every day starts out fifty-fifty."--'Adam Smith', The Money Game

In probability there is no such thing as a "hot hand." We would say that the chance of Kobe getting heads on the next flip is exactly 1/2, regardless of what his first five flips happened to be.

In theory, sports could be different from coin flips. If your mechanics are different on different days, then your probability of success will be different on different days. Therefore, successive basketball shots or baseball at-bats might not be independent from one another.

In practice, statisticians who have looked at basketball and baseball have found that the "hot hand" is largely an illusion. The fact that a basketball player has made a large number of shots in a row does not increase the probability that he will make the next one. In baseball, the statistical probability of a player getting a hit when he is "cold" usually turns out to be the same as when he is "hot."

Statistically, the assumption that each at-bat or basketball shot is independent of previous events holds up fairly well. The "law of averages" and the "hot hand" are just superstitions.