In a binomial setting with $n$ independent Bernoulli trials, probability of success $p$, and failure probability $q=1-p$, the probability of exactly $r$ successes is
A${}^nC_r p^r q^{n-r}$
B${}^nC_r p^{n-r} q^r$
C$p^r+q^{n-r}$
D$\dfrac{p^r q^{n-r}}{{}^nC_r}$
Answer & Solution
Correct answer: A. ${}^nC_r p^r q^{n-r}$
For binomial distribution, the number of ways to choose which $r$ trials are successes is ${}^nC_r$. Each such outcome has probability $p^r q^{n-r}$. So the required probability is ${}^nC_r p^r q^{n-r}$. Option B reverses success and failure exponents.
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