Use the Poisson Distribution to find the indicated probability.
A computer salesman averages 0.9 sales per week. Use the Poisson distribution to find the probability that in a randomly selected week the number of computers sold is 1. (Round answer to 4 decimal places.)
To find the probability using the Poisson distribution, we can use the formula:
P(x) = (e^(-λ) * λ^x) / x!
where P(x) is the probability of x occurrences, λ (lambda) is the average or expected number of occurrences, e is the mathematical constant approximately equal to 2.71828, and x! denotes the factorial of x.
In this case, the average number of sales per week is λ = 0.9, and we want to find the probability of selling exactly 1 computer in a randomly selected week, so x = 1.
Plugging in these values into the formula, we have:
P(1) = (e^(-0.9) * 0.9^2) / 1!
To calculate this, follow these steps:
1. Calculate e^(-0.9): In most calculators, you can enter -0.9 and then press the exponential (e^x) key to find the value. The result is approximately 0.40657.
2. Calculate 0.9^1: This is simply 0.9 raised to the power of 1, which equals 0.9.
3. Calculate 1! (factorial of 1): The factorial of 1 is 1.
4. Substitute these values into the formula and calculate the probability:
P(1) = (0.40657 * 0.9) / 1
P(1) ≈ 0.36591
Therefore, the probability that the number of computers sold in a randomly selected week is 1, using the Poisson distribution, is approximately 0.3659 (rounded to 4 decimal places).