26% of college students say they use credit cards because of the rewards program. You randomly select 10 college students and ask each to name the reason he or she uses credit cards. Find the probability that the number of college students who say they use credit cards because of the rewards program is (a) exactly two, (b) more than two,
To find the probability of the number of college students who say they use credit cards because of the rewards program, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where
P(x) is the probability of exactly x successes,
n is the total number of trials,
x is the number of successes,
p is the probability of success in a single trial,
C(n, x) is the binomial coefficient.
Given that 26% of college students say they use credit cards because of the rewards program, we can assume that the probability of success (p) is 0.26. The probability of not using credit cards because of the rewards program (q) is 1 - p = 1 - 0.26 = 0.74.
(a) To find the probability that exactly two students say they use credit cards because of the rewards program, we substitute the values into the binomial probability formula:
P(2) = C(10, 2) * 0.26^2 * 0.74^(10-2)
C(10, 2) = 10! / (2! * (10-2)!) = 45
P(2) = 45 * 0.26^2 * 0.74^8 ≈ 0.2717
Therefore, the probability that exactly two college students say they use credit cards because of the rewards program is approximately 0.2717.
(b) To find the probability that more than two students say they use credit cards because of the rewards program, we need to calculate the probabilities for x = 3, 4, 5, 6, 7, 8, 9, and 10 and sum them up.
P(more than 2) = P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
Using the same formula as above, we can calculate the probabilities for each value of x:
P(3) = C(10, 3) * 0.26^3 * 0.74^7
P(4) = C(10, 4) * 0.26^4 * 0.74^6
P(5) = C(10, 5) * 0.26^5 * 0.74^5
P(6) = C(10, 6) * 0.26^6 * 0.74^4
P(7) = C(10, 7) * 0.26^7 * 0.74^3
P(8) = C(10, 8) * 0.26^8 * 0.74^2
P(9) = C(10, 9) * 0.26^9 * 0.74^1
P(10) = C(10, 10) * 0.26^10 * 0.74^0
Calculating each of these probabilities and summing them up will give the probability of getting more than two students who say they use credit cards because of the rewards program.
P(more than 2) = P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
Finally, calculate each of these probabilities and sum them up to find the probability of getting more than two students who say they use credit cards because of the rewards program.