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,
To find the probability that Mystery Pizza receives a certain number of telephone orders between 11 P.M. and midnight on Thursday night, we can use the Poisson distribution. The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space.
The formula for the Poisson distribution is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
P(x; λ) is the probability of getting exactly x events in an interval,
e is the mathematical constant approximately equal to 2.71828,
λ is the average number of events in the interval,
x is the actual number of events that result.
In this case, the average number of telephone orders per hour is 4.2. Since we want to find the probability of receiving a certain number of orders between 11 P.M. and midnight, we can assume that the interval is one hour.
Let's calculate the probability for a few different values of x.
For x = 0:
λ = 4.2
P(0; 4.2) = (e^(-4.2) * 4.2^0) / 0! = e^(-4.2) ≈ 0.0149
For x = 1:
λ = 4.2
P(1; 4.2) = (e^(-4.2) * 4.2^1) / 1! = 4.2 * e^(-4.2) ≈ 0.0622
For x = 2:
λ = 4.2
P(2; 4.2) = (e^(-4.2) * 4.2^2) / 2! = (4.2^2 * e^(-4.2)) / 2 ≈ 0.1303
You can continue this calculation for different values of x to find the probability for a specific number of telephone orders between 11 P.M. and midnight on Thursday night using the Poisson distribution formula.