$$ 26 \% $$ 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) $$ \mathrm{P}(2)=0.274 $$ (Round to the nearest thousandth as needed.) (b) $$ \mathrm{P}(\mathrm{x}2)=0.503 $$ (Round to the nearest thousandth as needed.) (c) $$ \mathrm{P}(2 \leq x \leq 5)=\square $$ (Round to the nearest thousandth as needed.)

Question

$$ 26 \% $$ 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) $$ \mathrm{P}(2)=0.274 $$ (Round to the nearest thousandth as needed.) (b) $$ \mathrm{P}(\mathrm{x}>2)=0.503 $$ (Round to the nearest thousandth as needed.) (c) $$ \mathrm{P}(2 \leq x \leq 5)=\square $$ (Round to the nearest thousandth as needed.)

UpStudy AI · Accepted Answer

To solve part (c) of the problem, we need to find the probability that the number of college students who say they use credit cards because of the rewards program is **between two and five inclusive**.

Given:
- Probability of a student using a credit card for rewards ($$ p $$) = 26% or 0.26
- Number of students selected ($$ n $$) = 10
- We are dealing with a **binomial distribution** since each trial (student) is independent and has two possible outcomes (uses credit cards for rewards or not).

### Part (c): $$ \mathrm{P}(2 \leq x \leq 5) $$

We need to calculate:
$$
\mathrm{P}(2 \leq X \leq 5) = \mathrm{P}(X=2) + \mathrm{P}(X=3) + \mathrm{P}(X=4) + \mathrm{P}(X=5)
$$

Given that parts (a) and (b) have already provided:
- $$ \mathrm{P}(X=2) = 0.274 $$
- $$ \mathrm{P}(X>2) = 0.503 $$

However, to ensure accuracy, let's calculate each probability individually using the binomial probability formula:
$$
\mathrm{P}(X=k) = \binom{n}{k} p^k (1-p)^{n-k}
$$
where $$ \binom{n}{k} $$ is the binomial coefficient.

#### 1. $$ \mathrm{P}(X=2) $$
$$
\mathrm{P}(X=2) = \binom{10}{2} (0.26)^2 (0.74)^8 \approx 0.274
$$

#### 2. $$ \mathrm{P}(X=3) $$
$$
\mathrm{P}(X=3) = \binom{10}{3} (0.26)^3 (0.74)^7 \approx 0.256
$$

#### 3. $$ \mathrm{P}(X=4) $$
$$
\mathrm{P}(X=4) = \binom{10}{4} (0.26)^4 (0.74)^6 \approx 0.157
$$

#### 4. $$ \mathrm{P}(X=5) $$
$$
\mathrm{P}(X=5) = \binom{10}{5} (0.26)^5 (0.74)^5 \approx 0.066
$$

### Summing Up the Probabilities
$$
\mathrm{P}(2 \leq X \leq 5) = 0.274 + 0.256 + 0.157 + 0.066 = 0.753
$$

**Final Answer:**
$$
\mathrm{P}(2 \leq X \leq 5) = 0.753
$$

Rounded to the nearest thousandth, the probability is **0.753**.
The probability that between two and five college students use credit cards because of rewards is 0.753.

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