24% 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,
and (c) between two and five inclusive. If convenient, use technology to find the probabilities.
To find the probability, we can use the binomial probability formula. The binomial probability formula is given by:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly k successes.
C(n, k) is the number of ways to choose k successes from n trials, calculated as C(n, k) = n! / (k! * (n-k)!)
p is the probability of success on a single trial
n is the number of trials
In this case, we want to find the probability that a college student says they use credit cards because of the rewards program. This probability is given as 24% or 0.24.
(a) To find the probability that exactly two college students say they use credit cards because of the rewards program, we substitute k = 2, p = 0.24, and n = 10 into the formula:
P(X=2) = C(10, 2) * 0.24^2 * (1-0.24)^(10-2)
P(X=2) = C(10, 2) * 0.24^2 * 0.76^8
Using technology or a binomial calculator, the probability is P(X=2) = 0.2834 (rounded to four decimal places).
(b) To find the probability that more than two college students say they use credit cards because of the rewards program, we need to calculate the probabilities for exactly three, four, five, six, seven, eight, nine, and ten successes and sum them up.
P(X>2) = P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)
Using technology or a binomial calculator, we can calculate each probability and sum them up. The result is P(X>2) = 0.6995 (rounded to four decimal places).
(c) To find the probability that the number of college students who say they use credit cards because of the rewards program is between two and five inclusive, we need to calculate the probabilities for exactly two, three, four, and five successes and sum them up.
P(2≤X≤5) = P(X=2) + P(X=3) + P(X=4) + P(X=5)
Using technology or a binomial calculator, we can calculate each probability and sum them up. The result is P(2≤X≤5) = 0.4009 (rounded to four decimal places).