The number of calls received by a car towing service averages 14.4 per day (per 24 hour period). After finding the mean number of calls per hour, find the probability that in a randomly selected hour the number of calls is 0
To find the mean number of calls per hour, we need to divide the average number of calls per day by the number of hours in a day. Since there are 24 hours in a day, we can calculate the mean number of calls per hour as follows:
Mean Number of Calls per Hour = Average Number of Calls per Day / Number of Hours in a Day
Mean Number of Calls per Hour = 14.4 / 24
Mean Number of Calls per Hour = 0.6
Now, to find the probability that in a randomly selected hour the number of calls is 0, we can use the Poisson distribution. The Poisson distribution is commonly used to model the number of events (such as phone calls) that occur in a fixed interval of time.
The probability mass function (PMF) for the Poisson distribution is given by:
P(X=k) = (e^(-λ) * λ^k) / k!
Where:
P(X=k) is the probability of getting exactly k events,
e is the mathematical constant (approximately 2.71828),
λ is the mean number of events per interval, and
k is the actual number of events.
In this case, since we want to find the probability of getting 0 calls in a randomly selected hour, k would be 0, and λ would be 0.6 (the mean number of calls per hour we calculated earlier).
Let's calculate the probability using this formula:
P(X=0) = (e^(-0.6) * 0.6^0) / 0!
The term 0! (0 factorial) is equal to 1, so the equation simplifies to:
P(X=0) = e^(-0.6)
Now, we can calculate the probability using the value of e = 2.71828:
P(X=0) = 2.71828^(-0.6)
P(X=0) ≈ 0.5488
Therefore, the probability that in a randomly selected hour there will be 0 calls is approximately 0.5488.