22​% 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.
Question content area bottom
Part 1
​(a) ​P(2)equals
  
0.298 ​(Round to the nearest thousandth as​ needed.) 
Part 2
​(b) ​P(xgreater than​2)equals
  
enter your response here ​(Round to the nearest thousandth as​ needed.) 

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.

22% 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.

Learn how to solve a binomial probability problem involving college students' reasons for using credit cards. The problem covers calculating the probability of exactly, more than, and between a given number of students using credit cards due to rewards programs. Key concepts include binomial distribution and cumulative probability. Suitable for Grades 10-12.

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