Final answer:
The probability that fewer than two out of 15 patients will experience side effects from a certain drug is found using the binomial probability formula. By calculating the probabilities for exactly zero and one patient having side effects and then summing these probabilities, we determine the correct answer to be 0.288.
Step-by-step explanation:
The question is asking for the probability that fewer than two out of 15 patients will have a side effect from a certain drug, given that the drug causes side effects in 12% of patients. This is a binomial probability problem where we need to find P(X < 2) for X being the random variable representing the number of patients with side effects.
First, we calculate the probability of exactly one patient having a side effect, and the probability of no patients having a side effect. The binomial probability formula is P(X = k) = (n choose k)(p^k)(1-p)^(n-k), where 'n' is the number of trials (patients), 'k' is the number of successes (side effects), and 'p' is the probability of success (side effect).
- Probability of no patients (k=0) having a side effect: P(X = 0) = (15 choose 0)(0.12^0)(0.88^15)
- Probability of exactly one patient (k=1) having a side effect: P(X = 1) = (15 choose 1)(0.12^1)(0.88^14)
By adding these two probabilities, we get the total probability of fewer than two patients having side effects:
P(X < 2) = P(X = 0) + P(X = 1).
After calculating the probabilities for k=0 and k=1, we add them to find the final probability which corresponds to answer choice (b) 0.288.