Between 11 P.M. and midnight on Thursday night, Mystery Pizza gets an average of 5.7 telephone orders per hour.
a) Find the probability that at least 32 minutes will elapse before the next telephone order.
b) Find the probability that less than 17 minutes will elapse.
c) Find the probability that between 17 and 32 minutes will elapse
To solve these problems, we need to convert the given average into a rate per minute. We know that the average number of telephone orders per hour is 5.7, so dividing this by 60 (the number of minutes in an hour) will give us the average rate per minute.
a) Find the probability that at least 32 minutes will elapse before the next telephone order:
To find this probability, we need to use the exponential distribution formula:
P(X > t) = e^(-λt)
where λ is the average rate per minute and t is the time interval in minutes.
In this case, λ = 5.7/60 (average rate per minute) and t = 32 minutes. Plugging in these values into the formula:
P(X > 32) = e^(-(5.7/60) * 32)
Using a calculator, we can evaluate this expression to find the probability.
b) Find the probability that less than 17 minutes will elapse:
To find this probability, we can use the cumulative distribution function of the exponential distribution:
P(X < t) = 1 - e^(-λt)
In this case, λ = 5.7/60 (average rate per minute) and t = 17 minutes. Plugging in these values into the formula:
P(X < 17) = 1 - e^(-(5.7/60) * 17)
Using a calculator, we can evaluate this expression to find the probability.
c) Find the probability that between 17 and 32 minutes will elapse:
To find this probability, we can subtract the probability of less than 17 minutes from the probability of at least 32 minutes:
P(17 < X < 32) = P(X > 17) - P(X > 32)
Using the same exponential distribution formula and the given values, we can calculate these probabilities with a calculator.
Note: Make sure to check if your calculator requires the probabilities to be entered as a decimal or a percentage.