It is estimated that two out of five high school students...

It is estimated that two out of five high school students would fall victim to a phishing e-mail (an online scam asking for sensitive information) if it appears to originate from their high school main office. In a random sample of five high school students, what is the probability that exactly two fall victim to a phishing e-mail that appears to originate from their high school main office? (A) 0.4 (B) 1.0 (C) $\left(\begin{array}{l}5 \\ 2\end{array}\right)(0.4)^{2}(0.6)^{3}$ (D) $\left(\begin{array}{l}5 \\ 2\end{array}\right)(0.4)^{2}(0.6)^{3}$ (E) $(0.4)^{2}(0.6)^{3}$

It is estimated that two out of five high school students would fall victim to a phishing e-mail (an online scam asking for sensitive information) if it appears to originate from their high school main office.
In a random sample of five high school students, what is the probability that exactly two fall victim to a phishing e-mail that appears to originate from their high school main office?
(A) 0.4
(B) 1.0
(C) $\left(\begin{array}{l}5 \\ 2\end{array}\right)(0.4)^{2}(0.6)^{3}$
(D) $\left(\begin{array}{l}5 \\ 2\end{array}\right)(0.4)^{2}(0.6)^{3}$
(E) $(0.4)^{2}(0.6)^{3}$

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Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes (often termed success and failure) and a constant probability of success. The probability of exactly k successes in n trials is given by the formula P(X=k) = (n choose k) × (p)^k × (1-p)^(n-k).

Combinatorial Analysis

Combinatorial analysis, specifically the concept of combinations, is used to determine the number of ways to choose k successes from n trials without regard to the order. This is calculated using the binomial coefficient, often represented as (n choose k), which plays a crucial role in computing the probability in binomial distribution scenarios.

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