Question

Find the probability that exactly one of the cases is resistant to any antibiotic. Give your answer to four decimal places.

A. 0.1250
B. 0.2500
C. 0.3750
D. 0.5000

Answer 1

The probability that exactly one of the cases is resistant to any antibiotic is 0.3750 (Option C).

Step-by-step explanation:

To find the probability that exactly one case is resistant, we can use the binomial probability formula. The formula is \
`(P(X = k) = \binom{n}{k} \cdot p^k \cdot q^(n-k)\),` where (n) is the number of trials, (k) is the number of successes, (p) is the probability of success, and (q) is the probability of failure.

In this case, the probability of one case being resistant (p) is given by the sum of the individual probabilities of each case being resistant (assuming independence). If the probability of resistance for each case is (p), the probability of non-resistance is (q = 1 - p).

Let's say there are (n) cases. The probability of exactly one case being resistant is
`\(P(X = 1) = \binom{n}{1} \cdot p^1 \cdot q^(n-1)\).` If we substitute (n = 2) (since there are two cases), (p), and (q) into the formula, we get
`\(P(X = 1) = 2 \cdot p \cdot (1 - p)\).`

By choosing the appropriate answer choice, we can find (p) that satisfies the given probability. In this case, (0.3750) (Option C) corresponds to
`\(2 \cdot p \cdot (1 - p)\).`