What is the formula for the expected number of successes in a binomial experiment with $n$ trials and probability of success p?
Choose the correct forumla below.
A. $E(X)=\sqrt{n p(1-p)}$
B. $E(X)=p^n$
C. $E(X)=n p$
D. $E(X)=(1-p)^n(1-p)^n$
Answer (1)
The problem asks for the formula of expected number of successes in a binomial experiment.
The expected value E ( X ) in a binomial experiment is the product of the number of trials n and the probability of success p .
The formula is expressed as E ( X ) = n p .
Therefore, the correct answer is n p .
Explanation
Understanding Binomial Experiments In a binomial experiment, we perform n independent trials, each with a probability of success p . We want to find the expected number of successes, denoted as E ( X ) .
Defining Expected Value The expected value (or mean) of a binomial distribution is a measure of the average number of successes we expect to see in n trials. It's calculated by multiplying the number of trials by the probability of success in each trial.
Stating the Formula The formula for the expected value of a binomial distribution is given by: E ( X ) = n p where:
n is the number of trials
p is the probability of success on each trial
Comparing with Options Now, let's compare this formula with the given options: A. E ( X ) = n p ( 1 − p ) (This is the standard deviation, not the expected value) B. E ( X ) = p n (This is the probability of n successes in n trials, not the expected value) C. E ( X ) = n p (This matches the formula for the expected value) D. E ( X ) = ( 1 − p ) n ( 1 − p ) n (This is not a standard formula related to binomial distribution)
Conclusion Therefore, the correct formula for the expected number of successes in a binomial experiment is: E ( X ) = n p
Examples
Consider a basketball player who takes 10 shots (trials) and has a 70% (0.7) chance of making each shot (probability of success). The expected number of successful shots is calculated as E ( X ) = n ∗ p = 10 ∗ 0.7 = 7 . This means, on average, the player is expected to make 7 out of 10 shots. This concept is useful in sports analytics, quality control, and other fields where repeated independent trials occur.
