Between 11 P.M. and midnight on Thursday night, Mystery Pizza gets an average of 4.2 telephone orders per hour.

Find the probability that,
a.
at least 30 minutes will elapse before the next telephone order

b.
less than 15 minutes will elapse

c.
between 15 and 30 minutes will elapse

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To find the probability for each scenario, we need to convert the time intervals into the corresponding number of orders. Given that Mystery Pizza gets an average of 4.2 telephone orders per hour, divide this by 60 to find how many orders they receive on average per minute.

a. Probability that at least 30 minutes will elapse before the next telephone order:
30 minutes is equivalent to 1800 seconds. Since the average number of orders per minute is 4.2/60, we can calculate the probability using the Poisson distribution with a rate parameter of 4.2/60 and an interval of 1800 seconds using the following formula:

P(X ≥ k) = 1 - P(X < k)

Where X is a random variable representing the number of orders in a given time interval, and k is the number of orders we are interested in.

b. Probability that less than 15 minutes will elapse:
15 minutes is equivalent to 900 seconds. Using the same formula as above, we can find the probability:

P(X < k)

c. Probability that between 15 and 30 minutes will elapse:
We need to find the probability of the time interval being between 900 and 1800 seconds. Again, using the formula mentioned earlier, we calculate:

P(k1 < X < k2) = P(X < k2) - P(X < k1)

Where k1 and k2 are the numbers of orders corresponding to the respective time intervals.

Now, we can input the relevant values and calculate the probabilities using the Poisson distribution formula.