Question

During certain time periods at a​ hospital, patients arriving at the emergency room have waiting times that are uniformly distributed between 0 and 7 minutes. Assume that a patient is randomly​ selected, and find the probability that the waiting time is less than 0.75 minutes. (Round to three decimal places)

Answer 1

Since the waiting times are uniformly distributed between 0 and 7 minutes, the probability density function (PDF) of the waiting time is:

f(x) = 1/7 for 0 ≤ x ≤ 7

0 otherwise

To find the probability that the waiting time is less than 0.75 minutes, we need to integrate the PDF from 0 to 0.75:

P(X < 0.75) = ∫[0,0.75] f(x) dx

P(X < 0.75) = ∫[0,0.75] 1/7 dx

P(X < 0.75) = [1/7 * x] [0,0.75]

P(X < 0.75) = 1/7 * 0.75

P(X < 0.75) = 0.1071

Therefore, the probability that the waiting time is less than 0.75 minutes is 0.1071 (rounded to three decimal places).