The times to process orders at the service counter of a pharmacy are exponentially distributed with mean 1 0 minutes. If 100 customers visit the counter in a 2-day period, what is the probability that at least half of them need to wait more than 10 minutes?

Therefore, the probability that at least half of them need to wait more than 10 minutes is 0.0031.

Step-by-step explanation:

The formula for the probability of an exponential distribution is:

P (x < b) = 1 - e^ (b/3)

Using the complement rule, we can determine the probability of a customer having to wait more than 10 minutes, by:

p = P (x > 10)

= 1 - P (x < 10)

= 1 - (1 - e^ (-10/10))

= e⁻¹

= 0.3679

The z-score is the difference in sample size and the population mean, divided by the standard deviation:

z = (p' - p) / √[p (1 - p) / n]

= (0.5 - 0.3679) / √[0.3679 (1 - 0.3679) / 100) ]

= 2.7393

Therefore, using the probability table, you find that the corresponding probability is:

P (p' ≥ 0.5) = P (z > 2.7393)

P (p' ≥ 0.5) = 0.0031

Therefore, the probability that at least half of them need to wait more than 10 minutes is 0.0031.