Question 1181224: A shoe store´s records show that 30% of the customers purchase by credit cards. This is 20 customers purchased shoes from the store.
a) Find the probability that at most 3 of the customers used a credit card.
b) What is the probability that at least 3 customers but not more than 6 used a credit card?
c) What is the expected number of customers to use credit cards?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this binomial probability problem:
**a) Probability that at most 3 customers used a credit card:**
This means we want the probability of 0, 1, 2, or 3 customers using a credit card. We'll use the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
* n = number of trials (customers) = 20
* x = number of successes (customers using credit card)
* p = probability of success (customer using credit card) = 0.3
* nCx = "n choose x" (the binomial coefficient)
We need to calculate P(0), P(1), P(2), and P(3) and then add them together.
* P(0) = (20C0) * (0.3)^0 * (0.7)^20 ≈ 0.0008
* P(1) = (20C1) * (0.3)^1 * (0.7)^19 ≈ 0.0068
* P(2) = (20C2) * (0.3)^2 * (0.7)^18 ≈ 0.0278
* P(3) = (20C3) * (0.3)^3 * (0.7)^17 ≈ 0.0716
P(at most 3) = P(0) + P(1) + P(2) + P(3) ≈ 0.0008 + 0.0068 + 0.0278 + 0.0716 ≈ 0.107
**b) Probability that at least 3 customers but not more than 6 used a credit card:**
This means we want the probability of 3, 4, 5, or 6 customers using a credit card. We already have P(3). Now we calculate P(4), P(5), and P(6):
* P(4) = (20C4) * (0.3)^4 * (0.7)^16 ≈ 0.1304
* P(5) = (20C5) * (0.3)^5 * (0.7)^15 ≈ 0.1789
* P(6) = (20C6) * (0.3)^6 * (0.7)^14 ≈ 0.1916
P(at least 3 but not more than 6) = P(3) + P(4) + P(5) + P(6) ≈ 0.0716 + 0.1304 + 0.1789 + 0.1916 ≈ 0.5725
**c) Expected number of customers to use credit cards:**
The expected value of a binomial distribution is given by:
E(x) = n * p
E(x) = 20 * 0.3
E(x) = 6
So, we expect 6 customers to use credit cards.
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