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. Define the Parameters for the Binomial Distribution

We define the parameters of the binomial distribution:

• Number of trials (n): 10 (the number of college students surveyed)
• Probability of success (p): 0.28 (the probability a college student uses credit cards for rewards)
• Probability of failure (q): $q = 1 - p = 0.72$

2. Calculate the Probability for Exactly Two Students (a)

Using the binomial probability formula:

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

For exactly two students ($k = 2$):

$$ P(X = 2) = \binom{10}{2} (0.28)^2 (0.72)^{10-2} $$

Calculate the components:

$$ \binom{10}{2} = \frac{10 \cdot 9}{2 \cdot 1} = 45 $$

And:

$$ P(X = 2) = 45 \cdot (0.28)^2 \cdot (0.72)^8 $$

3. Calculate the Probability for More Than Two Students (b)

To find the probability of more than two students, we calculate:

$$ P(X > 2) = 1 - P(X \leq 2) $$

Compute $P(X \leq 2)$ as:

$$ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) $$

Using the binomial probability formula for $k = 0$ and $k = 1$:

• For $k = 0$:

$$ P(X = 0) = \binom{10}{0} (0.28)^0 (0.72)^{10} $$

• For $k = 1$:

$$ P(X = 1) = \binom{10}{1} (0.28)^1 (0.72)^{9} $$

4. Calculate the Probability for Between Two and Five Students Inclusive (c)

For this condition, we compute:

$$ P(2 \leq X \leq 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) $$

Calculate each term:

• For $k = 3$:

$$ P(X = 3) = \binom{10}{3} (0.28)^3 (0.72)^{10-3} $$

• For $k = 4$:

$$ P(X = 4) = \binom{10}{4} (0.28)^4 (0.72)^{10-4} $$

• For $k = 5$:

$$ P(X = 5) = \binom{10}{5} (0.28)^5 (0.72)^{10-5} $$

5. Summarize the Calculated Probabilities

After calculating all necessary individual probabilities for each part (a, b, and c), we will sum them properly to get the final probabilities required.