A small college has 800 students, $10 \%$ of which are left-handed. Suppose they take an SRS of 8 students. Let $L=$ the number of left-handed students in the sample.

Which of the following would find $P(L=2)$ ?

Choose 1 answer:
(A) $\binom{800}{8}(0.10)^{2}(0.90)^{6}$
(B) $\binom{8}{2}(0.10)^{6}(0.90)^{2}$
(C) $\binom{8}{2}(0.10)^{2}(0.90)^{6}$
(D) $(0.10)^{6}(0.90)^{2}$
(E) $(0.10)^{2}(0.90)^{6}$

Question

A small college has 800 students, $10 \%$ of which are left-handed. Suppose they take an SRS of 8 students. Let $L=$ the number of left-handed students in the sample.

Which of the following would find $P(L=2)$ ?

Choose 1 answer:
(A) $\binom{800}{8}(0.10)^{2}(0.90)^{6}$
(B) $\binom{8}{2}(0.10)^{6}(0.90)^{2}$
(C) $\binom{8}{2}(0.10)^{2}(0.90)^{6}$
(D) $(0.10)^{6}(0.90)^{2}$
(E) $(0.10)^{2}(0.90)^{6}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the correct formula for the binomial probability distribution, which is $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $(1-p)$ is the probability of failure. ‖ ‖ step 2 ⋮ Apply the formula to the given problem where $n=8$, $k=2$, $p=0.10$, and $(1-p)=0.90$. ‖ ‖ step 3 ⋮ Calculate the binomial coefficient $\binom{8}{2}$, which represents the number of ways to choose 2 left-handed students from a sample of 8. ‖ ‖ step 4 ⋮ Multiply the binomial coefficient by $p^k$ which is $(0.10)^2$ and by $(1-p)^{n-k}$ which is $(0.90)^6$. ‖ ‖ step 5 ⋮ The correct expression for the probability of having exactly 2 left-handed students in the sample is $\binom{8}{2}(0.10)^2(0.90)^6$. ‖ ***C*** ∻Key Concept∻ ⚹Binomial Probability Distribution⚹ ∻Explanation∻ ⚹The binomial probability distribution formula is used to calculate the probability of having a certain number of successes in a fixed number of independent trials of a binary outcome.⚹◊What is the formula for calculating the probability of having exactly $ k $ successes in $ n $ trials using the binomial distribution⍭ Generate me a similar question◊

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