Question

What is the mathematical expectation of the number of heads when 3 fair coins are tossed?

Answer 1

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µ=E[T] = 0 (1/8) + 1 (3/8) + 2 (3/8) + 3 (1/8) = 12/8 = 3/2 = 1.5. This can be interpreted as the average number of heads per sequence of 3 tosses if the experiment is repeated a large number of times.

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Consider again the count of heads in 3 tosses of a fair coin.  If this experiment is repeated, say 10 times, and the number of heads in each series of 3 tosses is counted, you will have a set of numbers like 0,1,3,1,2,2,1,1,3,0.  The average of these numbers is the average value for the random variable, the number of heads in 3 tosses of a fair coin.  To see what happens in a larger number of runs of the experiment, again link here, and when the page opens click the red die in front of number 4.  Set the number of coins at 3 and run the experiment 100 times.  What is the average number of heads per 3 tosses?  Now reset and run the experiment 1000 times.  What is the average number of heads per 3 tosses?

The long-term average number of heads is called the expected value of the random variable, the number of heads in 3 tosses of a fair coin.  This expected value can be found for most random variables.  Think of expected value as the average value of a random variable.

There is an easier way to find the expected value of this (or any) discrete random variable.  If the experiment of tossing the coin 3 times is repeated for a large number, N, times, the experiment will end in 0 heads n0 times, in 1 head n1 times, in 2 heads n2 times, and in 3 heads n3 times.  The total number of heads is 0 n0 + 1 n1 + 2 n2 + 3 n3, and the average number of heads per run of the experiment is

• (0 n0 + 1 n1 + 2 n2 + 3 n3)/N = 0 (n0/N) + 1 (n1/N) + 2 (n2/N) + 3 (n3/N)

For large N, (n0/N) ~ P[0 Heads], (n1/N) ~ P[1 Head], (n2/N) ~ P[2 Heads], (n3/N) ~ P[3 Heads], so the average number of heads per run of the experiment is

• 0 P[0 Heads] + 1P[1 Head] + 2 P[2 Heads] + 3 P[3 Heads]

This is called the Expected Value or Mean and is denoted, for a general random variable X, by E[X].  It can be computed by

The Binomial Random Variable

1. Definition--A binomial random variable with parameters n and p is a count of the number of successes in n experiments (or trials).
2. Each trial can result in only two outcomes, a success, S, or a failure, F.
3. The probability of a success on any trial is P[S] = p and the probability of a failure on any trial is P[F] = 1-p = q
4. The outcome of any trial has no effect on outcomes of other trials.
5. The binomial random variable is a count of the number of successes in n trials.
6. An example is the toss of a fair coin 3 times.  If you think of a success as a head, the count of the number of heads in 3 tosses satisfies the definition of a binomial random variable.  Here n=3 and p=1/2.