31% 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) $p(2) = 0.222$ (round to the nearest thousandth as needed.)
(b) $p(x 2) = 0.644$ (round to the nearest thousandth as needed.)
(c) $p(2 \le x \le 5) = $ (round to the nearest thousandth as needed.)

Question

31% 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) \$p(2) = 0.222\$ (round to the nearest thousandth as needed.)
(b) \$p(x > 2) = 0.644\$ (round to the nearest thousandth as needed.)
(c) \$p(2 \le x \le 5) = \$ (round to the nearest thousandth as needed.)

(a) $p(2) = 0.222$ (round to the nearest thousandth as needed.)
(b) $p(x > 2) = 0.644$ (round to the nearest thousandth as needed.)
(c) $p(2 \le x \le 5) = $ (round to the nearest thousandth as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Probability Formula", "Cumulative Binomial Probability" ], "new_concepts": [], "current_concepts": [ "Binomial Probability Formula", "Cumulative Binomial Probability" ] } ## Define binomial parameters $$ n = 10, \quad p = 0.31, \quad q = 1 - p = 0.69 $$ ## Calculate individual probabilities for $x = 2, 3, 4, 5$ $$ \begin{aligned} P(2) &= \binom{10}{2} (0.31)^2 (0.69)^8 \approx 45 \times 0.0961 \times 0.05138 \approx 0.2222 \ P(3) &= \binom{10}{3} (0.31)^3 (0.69)^7 \approx 120 \times 0.02979 \times 0.07447 \approx 0.2662 \ P(4) &= \binom{10}{4} (0.31)^4 (0.69)^6 \approx 210 \times 0.00924 \times 0.1079 \approx 0.2094 \ P(5) &= \binom{10}{5} (0.31)^5 (0.69)^5 \approx 252 \times 0.00286 \times 0.1564 \approx 0.1129 \end{aligned} $$ ## Sum the probabilities for the interval $2 \le x \le 5$ $$ \begin{aligned} P(2 \le x \le 5) &= P(2) + P(3) + P(4) + P(5) \ &\approx 0.2222 + 0.2662 + 0.2094 + 0.1129 \ &= 0.8107 \approx 0.811 \end{aligned} $$ (c) $P(2 \le x \le 5) =$ 0.811 (Round to the nearest thousandth as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Cumulative Binomial Probability" ] }

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Question

Explanation:

Define binomial parameters

Calculate individual probabilities for \(x = 2, 3, 4, 5\)

Sum the probabilities for the interval \(2 \le x \le 5\)

Answer: