31% 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.

​(a) ​P(2)equals=

This problem involves a **binomial probability distribution** because there are only two possible outcomes for each student: they either use credit cards for the rewards program (success) or they do not (failure). The key components of a binomial distribution are:

- **n (number of trials)**: 10 (you select 10 students)
- **p (probability of success)**: 0.31 (31% of students use credit cards for rewards)
- **q (probability of failure)**: \(1 - p = 0.69\)
- **x (number of successes)**: the value of \( x \) will vary based on the specific question.

### (a) Probability that exactly two students say they use credit cards because of the rewards program:

The formula for binomial probability is:

\[
P(x) = \binom{n}{x} p^x q^{n-x}
\]

Where:
- \( \binom{n}{x} \) is the binomial coefficient, which calculates the number of ways to choose \( x \) successes from \( n \) trials. It is calculated as:

\[
\binom{n}{x} = \frac{n!}{x!(n-x)!}
\]

Now, for \( x = 2 \):

\[
P(2) = \binom{10}{2} (0.31)^2 (0.69)^8
\]

I'll calculate this for you.The probability that exactly two college students out of 10 say they use credit cards because of the rewards program is approximately \( P(2) = 0.222 \), or 22.2%.

Would you like to move forward with parts (b) and (c) of the problem, or have any other questions about this result?

Here are 5 follow-up questions to consider:

1. What is the probability that none of the students use credit cards for rewards?
2. How does the binomial distribution change if the sample size increases?
3. How do you calculate the mean and variance of a binomial distribution?
4. What if the probability of using credit cards for rewards was 50% instead of 31%?
5. How can this problem be solved using a normal approximation for large samples?

**Tip**: When solving binomial problems, it's often helpful to use technology (like a calculator or software) for faster calculations, especially for large sample sizes.

31% 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.

Explore how to solve a binomial probability problem using the example of 31% of college students who use credit cards for rewards. Learn to calculate the probability that exactly two, more than two, and between two and five students choose rewards as the reason for using credit cards. Suitable for high school students in Grades 10-12.

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