The survival rate during a risky operation for patients with no other hope of survival is 87%. What is the probability that exactly four of the next five patients survive this operation? (Give your answer correct to three decimal places.)
I got 3.48 as the answer
To find the probability that exactly four out of the next five patients survive the operation, we can use the binomial probability formula.
The binomial probability formula states that the probability of getting exactly k successes in n independent Bernoulli trials, each with a probability of success p, is given by:
P(k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case, we want to find the probability that exactly four out of five patients survive the operation. The survival rate is given as 87%, which means the probability of success (p) is 0.87. The number of trials (n) is 5, and we want exactly four successes (k).
Using the formula, we have:
P(4) = (5 choose 4) * 0.87^4 * (1 - 0.87)^(5 - 4)
(5 choose 4) = 5! / (4! * (5 - 4)!) = 5
P(4) = 5 * 0.87^4 * (1 - 0.87)^1
Calculating this expression gives:
P(4) = 5 * 0.87^4 * 0.13
P(4) = 0.366
Therefore, the probability that exactly four out of the next five patients survive the operation is 0.366 (or 0.366 in decimal notation).