The number of phone calls per five minutes in an office has a mean of six.(a) What is the probability there will be exactly five calls in a five-minute period?
Question
The number of phone calls per five minutes in an office has a mean of six.(a) What is the probability there will be exactly five calls in a five-minute period?
Solution
To find the probability of exactly five phone calls in a five-minute period, we can use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events occurring 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 x events occurring in the given interval
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average number of events in the given interval
- x is the number of events we are interested in
In this case, the mean number of phone calls per five minutes is given as six, so λ = 6. We want to find the probability of exactly five calls, so x = 5.
Plugging these values into the formula, we get:
P(5; 6) = (e^(-6) * 6^5) / 5!
Now, we can calculate this probability step by step:
Step 1: Calculate e^(-6)
- e^(-6) is approximately 0.00248
Step 2: Calculate 6^5
- 6^5 is equal to 7776
Step 3: Calculate 5!
- 5! (5 factorial) is equal to 5 * 4 * 3 * 2 * 1 = 120
Step 4: Plug the values into the formula
- P(5; 6) = (0.00248 * 7776) / 120
- P(5; 6) is approximately 0.1606
Therefore, the probability of exactly five phone calls in a five-minute period is approximately 0.1606, or 16.06%.
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