From experience, the manager of a bookstore knows that 40% of the people browsing in his store will make a purchase. If on a certain day, 360 people come into the store

a) what is the probability that exactly 144 of them make a purchase?

b) what is the probability that at more 120 make a purchase?

To find the probability of these events, we can use the binomial probability formula. The formula to calculate the probability of exactly "k" successes in "n" trials is:

P(k) = (nCk) * p^k * (1 - p)^(n - k)

Where:
- P(k) is the probability of exactly "k" successes
- n is the number of trials (total number of people in this case)
- k is the number of successes (people making a purchase in this case)
- p is the probability of success (the probability of a given person making a purchase)
- nCk represents the binomial coefficient, also known as "n choose k" (the number of possible combinations of selecting "k" successes from "n" trials)

Now let's calculate the probabilities for the given scenarios:

a) What is the probability that exactly 144 of them make a purchase?

- n = 360 (total number of people)
- k = 144 (number of successes)
- p = 0.40 (probability of a person making a purchase)

Using the binomial probability formula, we get:

P(144) = (360C144) * (0.40^144) * (0.60^(360-144))

To evaluate this, we need to calculate the binomial coefficient (360C144). This can be calculated as:

(360C144) = 360! / (144! * (360-144)!)

Now, substitute the values and calculate the probability P(144).

b) What is the probability that at least 120 make a purchase?

To find the probability that at least 120 people make a purchase, we need to calculate the sum of probabilities for all possible values of successes from 120 to 360. We can use the cumulative distribution function (CDF) to calculate this.

P(at least 120) = P(120) + P(121) + ... + P(360)

For each value of "k" (120 to 360), calculate the probabilities P(k) using the binomial probability formula. Finally, sum up all these probabilities to get the probability of at least 120 people making a purchase.

I will now calculate these probabilities for you.