Question

A student is taking a multiple-choice exam which has 7 test questions and each has 4 options. Assume that the student has no knowledge of the subject material but guesses the correct answer with a probability of 0.75. What is the probability that she will answer at most 3 questions correctly?

Answer 1

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Answer 2

Answer:

The probability that she will answer at most 3 questions correctly is 0.0755

Step-by-step explanation:

• The concept used is the binomial theorem.
• Binomial theorem is used to expand the terms.
• Number of questions=7
• Probability of correct answer=0.75
• Probability of wrong answer=1-0.75

                                                       =0.25

• The probability that she will answer at most 3 questions correctly is given as

        $P=^nC_{r}p^rq^{n-r} \\\\P=^7C_{r}(0.75)^r (0.25)^{7-r} \ \ \ \ \ \ where\ r\ range\ from (0,1,2,3)\\\\P=^7C_{0}(0.75)^0 (0.25)^{7} +^7C_{1}(0.75)^1 (0.25)^{7-1} +^7C_{2}(0.75)^2 (0.25)^{7-2} +^7C_{3}(0.75)^3 (0.25)^{7-3}\\\\ P=^7C_{0}(0.75)^0 (0.25)^{7} +^7C_{1}(0.75)^1 (0.25)^{6} +^7C_{2}(0.75)^2 (0.25)^{5} +^7C_{3}(0.75)^3 (0.25)^{4}\\\\P=(0.25)^{7} +7(0.75)^1 (0.25)^{6} +21(0.75)^2 (0.25)^{5} +^35(0.75)^3 (0.25)^{4}\\P=0.0755$