This is a binomial probability problem, where we are interested in the probability that a certain number of college students, out of a random sample, say they use credit cards because of the rewards program.

Given:

• Probability $p = 0.22$ (22% of students use credit cards due to rewards)
• $n = 10$ (you select 10 students)
• Let $X$ be the number of students who use credit cards for rewards. Thus, $X \sim \text{Binomial}(n=10, p=0.22)$.

The binomial probability formula is: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ where $\binom{n}{k}$ is the binomial coefficient.

Let's solve for each part:

(a) Probability of exactly 2 students saying they use credit cards for rewards: $P(X = 2)$

We are given the result: $P(X = 2) = 0.298$ (Rounded to the nearest thousandth).

(b) Probability that more than 2 students say they use credit cards for rewards: $P(X > 2)$

To calculate this, we find: $P(X > 2) = 1 - P(X \leq 2)$ where $P(X \leq 2)$ is the cumulative probability for $X = 0, 1, 2$.

Let's calculate $P(X > 2)$.The probability that more than 2 college students say they use credit cards because of the rewards program is approximately $P(X > 2) = 0.383$ (rounded to the nearest thousandth).

Would you like details or have any questions? Here are five related questions for further practice:

1. What is the probability that no students use credit cards for the rewards program?
2. How would the probabilities change if you selected 20 students instead of 10?
3. What is the expected number of students out of 10 who use credit cards for rewards?
4. How would you calculate the variance of the binomial distribution in this context?
5. What is the probability that exactly 5 students use credit cards for rewards?

Tip: In binomial problems, calculating cumulative probabilities $P(X \leq k)$ helps solve for probabilities involving "less than" or "greater than" scenarios.