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.
Oh, college students and their credit cards! Let's dive into the probabilities with a little comedic twist, shall we?
(a) To find the probability of exactly two students saying they use credit cards because of the rewards program, we can use the binomial probability formula. Since we have a 24% chance for each student, we can calculate it as follows:
P(exactly two) = (10 choose 2) * (0.24^2) * (0.76^8)
Using a bit of magic (or just a calculator), we find that P(exactly two) is approximately 0.286.
(b) Now, let's aim for more! To find the probability of more than two students, we'll have to add up the probabilities for exactly three, four, five, six, seven, eight, nine, and ten students. Trust me, it'll be worth it!
P(more than two) = P(exactly three) + P(exactly four) + P(exactly five) + P(exactly six) + P(exactly seven) + P(exactly eight) + P(exactly nine) + P(exactly ten)
Using some fancy calculations, we find that P(more than two) is approximately 0.597.
(c) Lastly, we need to calculate the probability of having two, three, four, or five students who use credit cards because of rewards. Let me get my counting hat on!
P(between two and five inclusive) = P(exactly two) + P(exactly three) + P(exactly four) + P(exactly five)
After crunching some numbers, we find that P(between two and five inclusive) is approximately 0.816.
Remember, these numbers are just probabilities, and it's always tricky predicting human behavior, especially when it comes to credit cards. But hey, at least we had some fun along the way!
To find the probabilities, we can use the binomial probability formula.
The probability of getting exactly k successes in n trials, given a probability of success p, is given by:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where (n choose k) is the number of ways to choose k items from a set of n items, and is calculated as:
(n choose k) = n! / (k! * (n - k)!)
Let's calculate the probabilities using the provided information.
(a) To find the probability that exactly two college students say they use credit cards because of the rewards program:
n = 10 (number of trials)
k = 2 (number of successes)
p = 0.24 (probability of success)
Using the formula:
P(X = 2) = (10 choose 2) * (0.24)^2 * (1 - 0.24)^(10 - 2)
Using a calculator or software, we get:
P(X = 2) ≈ 0.28096
(b) To find the probability that more than two college students say they use credit cards because of the rewards program, we need to find the cumulative probability of getting 3, 4, 5, 6, 7, 8, 9, or 10 successes.
P(X > 2) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
Using a calculator or software, we can calculate the individual probabilities and subtract from 1:
P(X > 2) ≈ 0.8626
(c) To find the probability that between two and five college students (inclusive) say they use credit cards because of the rewards program, we need to find the cumulative probability of getting 2, 3, 4, or 5 successes.
P(2 ≤ X ≤ 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Again, using a calculator or software, we can calculate the individual probabilities and sum them up:
P(2 ≤ X ≤ 5) ≈ 0.7791
Therefore, the probabilities are:
(a) P(X = 2) ≈ 0.28096
(b) P(X > 2) ≈ 0.8626
(c) P(2 ≤ X ≤ 5) ≈ 0.7791
To find the probability in this scenario, we can use the binomial probability formula. The formula is:
P(x) = nCx * p^x * q^(n-x),
where
P(x) is the probability of exactly x successes,
n is the number of trials,
p is the probability of success on a single trial, and
q is the probability of failure on a single trial.
In this case:
- n is the number of college students you randomly selected, which is 10.
- p is the probability of a student using credit cards because of the rewards program, which is 24% or 0.24.
- q is the probability of a student not using credit cards because of the rewards program, which is 100% minus the 24%, or 76% or 0.76.
Now, let's calculate the probabilities:
(a) Probability of exactly two students saying they use credit cards because of the rewards program:
P(2) = 10C2 * 0.24^2 * 0.76^(10-2)
Using technology or a calculator, we can calculate the value:
P(2) ≈ 0.255
(b) Probability of more than two students saying they use credit cards because of the rewards program:
P(x > 2) = P(3) + P(4) + P(5) + ... + P(10)
To calculate this cumulative probability, you can add the individual probabilities for each value from 3 to 10. Using technology or a calculator, we get:
P(x > 2) ≈ 0.947
(c) Probability of between two and five students saying they use credit cards because of the rewards program (inclusive):
P(2 ≤ x ≤ 5) = P(2) + P(3) + P(4) + P(5)
Again, you can calculate this cumulative probability by adding the individual probabilities for each value from 2 to 5. Using technology or a calculator, we get:
P(2 ≤ x ≤ 5) ≈ 0.979
Remember to always check if the calculations and assumptions are appropriate for the given problem before using the results.
binomial distribution
p = .24
1-p = .76
n = 10
P(2) = C(10,2).24^2 (.76^8)
but C(10,2) = 45
so
45 * .0576 * .111 = .288
P(1) = 10 * .24 * .76^9
= .203
so
answer to b is 1 - .288 - .203
c is same way but boring. you do it