Final answer:
The probability of turning away one or more customers is 0.53, the center's capacity will not be fully utilized with a probability of 0.47, capacity must be increased to 4 to have a less than 0.10 chance of turning customers away, and the calculation of mean and standard deviation requires further steps.
Step-by-step explanation:
To answer these questions, we need to calculate probabilities based on the given probability distribution for the number of customers per day at the car painting center. The values of x represent the number of customers, and f(x) represents the probability of x customers arriving.
- To find the probability that one or more customers will be turned away on a given day, we need to consider days when the number of customers is more than the center's capacity, which is two customers. Hence, we add up the probabilities for 3, 4, and 5 customers: P(x > 2) = f(3) + f(4) + f(5) = 0.19 + 0.12 + 0.22 = 0.53.
- For the probability that the center's capacity will not be fully utilized on a day, we look at days with 0, 1, or 2 customers: P(x < 3) = f(0) + f(1) + f(2) = 0.05 + 0.19 + 0.23 = 0.47.
- To increase capacity so the probability of turning a customer away is less than 0.10, we add up the probabilities starting from the highest number of customers until the total is just below 0.10. Starting from 5 customers and subtracting the probabilities successively, we find that the capacity must be increased to at least 4 customers. This can be seen as P(x > 3) = f(4) + f(5) = 0.12 + 0.22 = 0.34; P(x > 4) = f(5) = 0.22, and since 0.22 < 0.34 and is close to 0.10, increasing capacity to 4 meets the requirement.
- To find the mean (μ) and standard deviation (σ) of X, we calculate μ = E[X] = Σ xf(x) and σ = sqrt(Σ (x-μ)^2f(x)). The mean is μ = (0 × 0.05) + (1 × 0.19) + (2 × 0.23) + (3 × 0.19) + (4 × 0.12) + (5 × 0.22) = 2.68. To calculate standard deviation, we need the value of σ = sqrt(Σ (x-μ)^2f(x)), which requires a few more steps to compute each term and sum them up.