Twenty-eight percent 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... Twenty-eight percent 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. Read more

Steps to Solve

1. Identify the parameters of the binomial distribution

In this scenario, we have:

• The probability of success (a student using credit cards because of the rewards program) $p = 0.28$ (28%).
• The probability of failure $q = 1 - p = 0.72$.
• The number of trials (students asked) $n = 10$.

2. Define the binomial probability formula

The binomial probability formula is given by:

$$ P(X = k) = \binom{n}{k} p^k q^{n-k} $$

where:

• $P(X = k)$ is the probability of exactly $k$ successes,
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.

3. Calculate probabilities for the specific cases

• (a) Exactly two: Calculate $P(X = 2)$

Using the parameters from step 1 and plugging $k = 2$ into the formula: $$ P(X = 2) = \binom{10}{2} (0.28)^2 (0.72)^{8} $$

• (b) More than two: Calculate $P(X > 2)$

This can be calculated as: $$ P(X > 2) = 1 - P(X \leq 2) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) $$

For $P(X = 0)$ and $P(X = 1)$, repeat the same formula as in step 2.

• (c) Between two and five inclusive: Calculate $P(2 \leq X \leq 5)$

Sum the probabilities: $$ P(2 \leq X \leq 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) $$

Calculate each of these using the binomial probability formula.