The probability a car salesman sells a car to a customer is 0.05
Assuming the salesmen sees 14 customers in a week, what is the probability he sells less than 2 cars?
Write answer using three decimal places
prob( less than 2 cars)
= prob(sell none) + prob(sell 1)
=
To find the probability that the car salesman sells less than 2 cars, we can use the binomial probability formula. The formula is as follows:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the number of trials (in this case, the number of customers the salesman sees in a week),
- k is the number of successes (the number of cars sold),
- p is the probability of success (the probability a car salesman sells a car to a customer),
- (nCk) is the binomial coefficient, which represents the number of ways to choose k items out of n.
First, let's calculate the probability of selling less than 2 cars. We need to find the probability when k = 0 and when k = 1, and then sum them up.
P(X < 2) = P(X = 0) + P(X = 1)
Now, let's substitute the values into the formula.
For when k = 0:
P(X = 0) = (nC0) * p^0 * (1 - p)^(n - 0) = 1 * 1^0 * (1 - p)^(n) = (1 - p)^n
For when k = 1:
P(X = 1) = (nC1) * p^1 * (1 - p)^(n - 1) = (n! / ((n - 1)! * 1!)) * p * (1 - p)^(n - 1) = n * p * (1 - p)^(n - 1)
Now, let's substitute the given values, n = 14 and p = 0.05, into the equations.
For when k = 0:
P(X = 0) = (1 - 0.05)^14
For when k = 1:
P(X = 1) = 14 * 0.05 * (1 - 0.05)^(14 - 1)
Finally, let's calculate the probabilities.
P(X < 2) = (1 - 0.05)^14 + 14 * 0.05 * (1 - 0.05)^(14 - 1)
Calculating this using a calculator or software, the answer is approximately 0.508. Therefore, the probability the car salesman sells less than 2 cars is 0.508 (rounded to three decimal places).