In a clinic, 62% of patients are female. Fifteen patients are selected at random with replacement. What is the probability that at least 1 of the fifteen patients is male?
To find the probability that at least 1 of the fifteen patients is male, we can calculate the probability of the complementary event - that is, the probability that none of the fifteen patients is male - and then subtract it from 1.
The probability that a randomly selected patient is female is 62%, which means that the probability that a randomly selected patient is male is 100% - 62% = 38%.
Since each patient is selected at random with replacement (meaning that after each selection, the patient is put back into the population before the next selection), the probability of selecting a female patient each time is the same for each selection. So, the probability that none of the fifteen patients is male is (0.62)^15, which is the probability of selecting a female patient 15 times in a row.
Now, we can calculate the probability that at least 1 of the fifteen patients is male. To do this, we subtract the probability of the complementary event from 1:
Probability of at least 1 male patient = 1 - Probability of no male patient = 1 - (0.62)^15
Therefore, the probability that at least 1 of the fifteen patients is male is approximately 1 - (0.62)^15 = 0.993 (rounded to three decimal places), or 99.3%.