Multiple Choice
Which of the following graphs best represents the probability mass function of a binomial distribution with and ?
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5. Binomial Distribution & Discrete Random Variables
Binomial Distribution
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For the binomial distribution, which of the following expressions correctly represents the probability of obtaining exactly successes in independent trials, each with probability of success ?
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Verified step by step guidance1
Recall that the binomial distribution models the number of successes in \( n \) independent trials, each with success probability \( p \). The probability of exactly \( k \) successes is given by the binomial probability formula.
The formula involves three components: the number of ways to choose which \( k \) trials are successes, the probability of those \( k \) successes occurring, and the probability of the remaining \( n-k \) trials being failures.
The number of ways to choose \( k \) successes out of \( n \) trials is given by the binomial coefficient, which is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
The probability of \( k \) successes is \( p^k \), and the probability of \( n-k \) failures is \( (1-p)^{n-k} \).
Putting it all together, the probability of exactly \( k \) successes in \( n \) trials is:
\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} = \frac{n!}{k!(n-k)!} \cdot p^k \cdot (1-p)^{n-k}
\]
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