Question

A restaurant gets an average of 12 phone orders in a 2 hour time period. In order to find the probability that the restaurant will get exactly 2 phone orders in a 30 minute period using the Poisson distribution, what is the average number of phone orders per 30 minutes?

Answer 1

Answer:

The average number of phone calls in a 30 minute period is 3.

The probability to receive exactly 2 calls in that period is 0.224.

Step-by-step explanation:

If we are using the same Poisson distribution in the 2 hour period, then the average of phone orders in a reduced interval will be reduced according to that interval. Since 30 minutes is four times smaller than 2 hours, then the average number of phone orders per 30 minutes is 12 * 1/4 = 3. This can also be computed with a Rule of 3

120 minutes -------------> 12 orders

30 minutes ---------------> X orders

X = 30*12/120 = 3

Lets call Y the amount of phone orders received during a specific (random) 30 minute period. Since the average was 3, then Y has a Poisson distribution with parameter 3. The probability of Y being equal to 2 is

$P(Y=2) = \frac{e^{-3} 3^2 }{2!} = \frac{9}{2} \, e^{-3} = 0.224$

Thus, the probability to receive exactly 2 calls in a 30 minute period is 0.224.