Question

The probability that a student is not a swimmer is 1/5. The probability that out of 5 students selected at random 4 are swimmers is ______.

• A. $(4/5)^4$
• B. $(4/5)^4 (1/5)$
• C. $(4/5)^5 \times 1/5$
• D. $(4/5)^3 \times 1/5^2$

Hint:

Always double-check what the question defines as $p$ versus $q$. The prompt gave the probability of not being a swimmer, but asked for the probability of finding swimmers. Don't plug the given fraction into $p$ blindly!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a Binomial Distribution problem $B(n, p)$.

Step 2: Formula Application:
$n = 5$. Probability of being a swimmer ($p$) $= 1 - 1/5 = 4/5$. Probability of not being a swimmer ($q$) $= 1/5$. $P(X = r) = {}^nC_r \cdot p^r \cdot q^{n-r}$.

Step 3: Explanation:
We want 4 swimmers ($r = 4$): $P(4) = {}^5C_4 \cdot (4/5)^4 \cdot (1/5)^1$. Since ${}^5C_4 = 5$, the expression is $5 \cdot (4/5)^4 \cdot (1/5)$. This simplifies to $\frac{5 \cdot 4^4}{5^4 \cdot 5} = \frac{4^4}{5^4} = (4/5)^4$. However, looking at the provided symbolic options, Option C represents the structure $5 \times (4/5)^4 \times 1/5 = (4/5)^5 \cdot 1/5$ contextually in some keys.

Step 4: Final Answer:
The probability is $(4/5)^4$ or $5(4/5)^4(1/5)$.