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 Question:
This is a standard Binomial Distribution problem. We are selecting a sample size of 5 students and want to find the exact probability of getting exactly 4 "successes" (swimmers).

Step 2: Key Formula or Approach:
The binomial probability formula is:
$$P(X = r) = \binom{n}{r} p^r q^{n-r}$$
Where:
$n$ = total number of trials (students selected).
$r$ = number of desired successes (swimmers).
$p$ = probability of success on a single trial.
$q$ = probability of failure on a single trial ($q = 1 - p$).

Step 3: Detailed Explanation:
Let a "success" be defined as a student who is a swimmer.
We are given the probability that a student is not a swimmer is 1/5. Therefore, $q = 1/5$.
The probability that a student is a swimmer is $p = 1 - q = 1 - 1/5 = 4/5$.
We are selecting 5 students, so $n = 5$.
We want exactly 4 swimmers, so $r = 4$.
Plug these values into the binomial formula:
$$P(X = 4) = \binom{5}{4} \left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^{5-4}$$
Calculate the combination $\binom{5}{4}$, which is equal to 5.
$$P(X = 4) = 5 \times \left(\frac{4}{5}\right)^4 \times \left(\frac{1}{5}\right)^1$$
Notice that the integer $5$ and the fraction $\frac{1}{5}$ perfectly cancel each other out:
$$P(X = 4) = \left( 5 \times \frac{1}{5} \right) \times \left(\frac{4}{5}\right)^4$$
$$P(X = 4) = 1 \times \left(\frac{4}{5}\right)^4 = \left(\frac{4}{5}\right)^4$$

Step 4: Final Answer:
The probability simplifies to $(4/5)^4$, matching option (a).