A binomial distribution describes the probability of getting exactly x successes in n trials, given the probability of success in a single trial is p:
P(exactly x successes) = nCk * pk * (1-p)n-k
For example, if we flip a coin three times, the number of times it lands on heads has the following binomial distribution, where X = number of times it lands on heads and P(X) = probability associated with that outcome:
The expected value of a binomial distribution can be written as: n * p
Put simply, the expected value is the number of trials multiplied by the probability of success on each trial.
This tutorial provides an intuitive explanation as well as a mathematical explanation of how this formula for expected value is derived.
Consider the example of flipping a coin 10 times. How many times would you expect the coin to land on heads?
Since the coin is equally likely to land on either heads or tails on each flip, you would expect the coin to land on heads half of the time. This means you would expect the coin to land on heads 5 times out of 10.
We could come up with the same answer if we use the formula for expected value of n * p. In this case n = 10 flips, and p (probability of landing on heads) = 0.5.
Thus, the expected number of heads = n * p = 10 * 0.5 = 5.
Consider the example of rolling a dice 12 times. How many times would you expect the dice to land on the number 4?
Since the dice is equally likely to land on numbers 1 through 6 with each roll, you would expect the dice to land on the number 4 about 1 out of 6 times. Since you’re rolling the dice 12 times, you would expect it to land on the number 4 just 2 times.
We could come up with the same answer if we use the formula for expected value of n * p. In this case n = 12 rolls, and p (probability of landing on 4) = 1/6.
Thus, the expected number of 4′s = n * p = 12 * (1/6) = 2.
Mathematical Explanation (Proof)
By the definition of expected value, we can write:
We can the rewrite the following as:
We can then factor out np and cancel out the k‘s, which gives us:
The right hand side of the equation is equal to:
Then, since the following is true:
We can say that:
This concludes the proof that the expected value of the binomial distribution is equal to n * p.