From experience, the manager of Kramer's Book Mart knows that 30% 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 seven will make a purchase?(round answer to four decimal places)
To find the probability that at least seven out of ten people who are browsing in the store will make a purchase, we need to use the binomial probability formula.
The binomial probability formula is as follows:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability that exactly k successes occur
n is the total number of trials
k is the number of successful outcomes
p is the probability of success on a single trial
In this case, n (total number of trials) is 10, k (number of successful outcomes) ranges from 7 to 10, and p (probability of success) is 0.30 (since the manager knows that 30% of people make a purchase).
Now we can calculate the probability for each value of k (7, 8, 9, and 10) and sum them up to get the probability of at least seven people making a purchase.
P(X >= 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Let's calculate these probabilities step by step:
For k = 7:
P(X = 7) = (10 C 7) * (0.30)^7 * (1-0.30)^(10-7) = 0.2668
For k = 8:
P(X = 8) = (10 C 8) * (0.30)^8 * (1-0.30)^(10-8) = 0.1201
For k = 9:
P(X = 9) = (10 C 9) * (0.30)^9 * (1-0.30)^(10-9) = 0.0282
For k = 10:
P(X = 10) = (10 C 10) * (0.30)^10 * (1-0.30)^(10-10) = 0.0028
Now, sum up these probabilities to get the final answer:
P(X >= 7) = 0.2668 + 0.1201 + 0.0282 + 0.0028 = 0.4179
Therefore, the probability that among ten people who are browsing in the store, at least seven will make a purchase is approximately 0.4179 (rounded to four decimal places).