EXAMPLE 4.23

Number of heads in four tosses of a coin. What is the probability distribution of the discrete random variable X that counts the number of heads in four tosses of a coin? We can derive this distribution if we make two reasonable assumptions:

  • • The coin is balanced, so it is fair and each toss is equally likely to give H or T.

  • • The coin has no memory, so tosses are independent.

The outcome of four tosses is a sequence of heads and tails such as HTTH. There are 16 possible outcomes in all. Figure 4.6 lists these outcomes along with the value of X for each outcome. The multiplication rule for independent events tells us that, for example,

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Figure 4.6 Possible outcomes in four tosses of a coin, Example 4.23. The outcomes are arranged by the values of the random variable X, the number of heads.

Each of the 16 possible outcomes similarly has probability 1/16. That is, these outcomes are equally likely.

The number of heads X has possible values 0, 1, 2, 3, and 4. These values are not equally likely. As Figure 4.6 shows, there is only one way that X = 0 can occur: namely, when the outcome is TTTT. So

The event {X = 2} can occur in six different ways, so that

We can find the probability of each value of X from Figure 4.6 in the same way. Here is the result:

Value of X 0 1 2 3 4
Probability 0.0625 0.25 0.375 0.25 0.0625