2424% 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.
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"2424%"?
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 need to use the binomial distribution formula. The binomial distribution is used when there are two possible outcomes (success or failure) for each trial, and the probability of success remains the same for each trial.
Let's define the variables:
n = number of trials (in this case, 10 college students)
p = probability of success (24.24% or 0.2424)
x = number of successes (the number of college students who say they use credit cards because of the rewards program)
(a) We want to find the probability that exactly two students say they use credit cards because of the rewards program. This can be calculated using the binomial probability formula, which is:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
Where (nCx) represents the number of combinations of choosing x successes from n trials.
Using the formula:
P(X = 2) = (10C2) * (0.2424)^2 * (1-0.2424)^(10-2)
Now we can calculate it:
P(X = 2) = (10! / (2!(10-2)!)) * 0.2424^2 * (0.7576)^8
To calculate it using technology, you can use a scientific calculator or statistical software like Excel or Python to get the exact result.
(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, ..., 10, and sum them up.
P(X > 2) = P(X = 3) + P(X = 4) + P(X = 5) + ... + P(X = 10)
Using technology, you can calculate each of these probabilities and sum them up.
(c) Similarly, 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 x = 2, 3, 4, 5 and sum them up.
P(2 ≤ X ≤ 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Again, using technology, you can calculate each of these probabilities and sum them up.