From experience, the manager of Kramer's Book Mart knows that 40% of the people who are browsing in the store will make a purchase. What is the probability that among ten people who are browsing in the store, at least five will make a purchase? (Round your answer to four decimal places.)
To calculate the probability that at least five people will make a purchase among ten people browsing, we can use the binomial probability formula.
The binomial probability formula is:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes
- n is the number of trials (in this case, the number of people browsing = 10)
- k is the number of successful outcomes (in this case, the number of people making a purchase)
- p is the probability of success (in this case, 40% = 0.4)
- (n C k) is the combination formula, which calculates the number of ways to choose k items from a set of n items.
In this case, we need to calculate the probabilities for k = 5, 6, 7, 8, 9, and 10 (at least five purchases).
P(X ≥ 5) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)
To calculate each of these probabilities, we substitute the values into the binomial probability formula:
P(X=5) = (10 C 5) * (0.4)^5 * (1-0.4)^(10-5)
P(X=6) = (10 C 6) * (0.4)^6 * (1-0.4)^(10-6)
P(X=7) = (10 C 7) * (0.4)^7 * (1-0.4)^(10-7)
P(X=8) = (10 C 8) * (0.4)^8 * (1-0.4)^(10-8)
P(X=9) = (10 C 9) * (0.4)^9 * (1-0.4)^(10-9)
P(X=10) = (10 C 10) * (0.4)^10 * (1-0.4)^(10-10)
Now, let's calculate each probability step-by-step:
To find the probability that at least five out of ten people will make a purchase, we can use the binomial probability formula.
The binomial probability formula is given by:
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
- k is the number of successful outcomes
- p is the probability of success in one trial (in this case, the probability of making a purchase)
- (1-p) is the probability of failure in one trial
In this case, n = 10 (number of people browsing), p = 0.4 (probability of making a purchase), and we need to find the probability of at least 5 people making a purchase.
To find the probability of at least five people making a purchase, we need to calculate the probabilities for five, six, seven, eight, nine, and ten people making a purchase, and then sum them up.
P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Calculating each individual probability using the binomial probability formula:
P(X = 5) = 10C5 * 0.4^5 * (1-0.4)^(10-5)
P(X = 6) = 10C6 * 0.4^6 * (1-0.4)^(10-6)
P(X = 7) = 10C7 * 0.4^7 * (1-0.4)^(10-7)
P(X = 8) = 10C8 * 0.4^8 * (1-0.4)^(10-8)
P(X = 9) = 10C9 * 0.4^9 * (1-0.4)^(10-9)
P(X = 10) = 10C10 * 0.4^10 * (1-0.4)^(10-10)
Using the binomial coefficient (nCk) formula:
10C5 = 10! / (5! * (10-5)!)
10C6 = 10! / (6! * (10-6)!)
10C7 = 10! / (7! * (10-7)!)
10C8 = 10! / (8! * (10-8)!)
10C9 = 10! / (9! * (10-9)!)
10C10 = 10! / (10! * (10-10)!)
Calculating each individual probability and summing them up:
P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Finally, we round the answer to four decimal places.
prob of purchase = .4
prob of no purchase = .6
prob of at least 5 from 10 will make purchase
= prob(5will buy) + prob(6 will buy) + ..+ prob(10 will buy)
....
lots of arithmetic, I will do the prob 6 will buy
= C(10,6) (.4)^6 (.6)^4
= ..
What might be a shorter way is to exclude the cases of none, 1, 2 , 3 or 4 will buy
= 1 - (prob(0 will buy + prob(1 will buy ...+prob(4 will buy)