Final answer:
The probability of exactly 6 events occurring in a Poisson distribution with a mean of 7 (P(6) when μ = 7) is calculated using the Poisson probability formula. This calculation can be performed by hand or by using a statistical software or graphing calculator with the Poisson probability distribution function.
Step-by-step explanation:
To find the probability of exactly 6 events occurring (P(6)) when the mean number of events (μ) is 7 for a Poisson distribution, we can use the formula for the Poisson probability:
P(x; μ) = (e-μμx)/x!
Where:
- e is the base of the natural logarithm, approximately equal to 2.71828,
- μ is the mean number of occurrences,
- x is the actual number of occurrences,
- x! is the factorial of x.
Plugging the values into the Poisson formula:
P(6; 7) = (e-776)/6!
Calculating the factorial and powers, then multiplying and dividing as per the formula will give us P(6).
However, in practice, this calculation is often performed using statistical software or a graphing calculator with the capability to compute Poisson probabilities. For instance, using a calculator with the Poisson probability distribution function, we would enter the mean (7) and the value of x (6) to obtain the probability.