A class is taking a multiple choice test with 10 questions where each

kursval7z

kursval7z

Answered question

2021-11-13

A class is taking a multiple choice test with 10 questions where each question has four possible answers. Assume that the answer to any question is independent of that to any other answer. Robert has forgotten to study for this test, so he simply guesses for each question. 
What is the probability that Robert guesses exactly 8 of the questions correctly?
What is the probability that Robert gets an 80% or better? Another way of saying this is, what is the probability that he guesses 8 or more questions correctly? This would be the probability of 8 plus the probability of 9 plus the probability of 10.

Answer & Explanation

Mary Ramirez

Mary Ramirez

Beginner2021-11-14Added 19 answers

Step 1
The probability that R guesses exactly 8 of the questions correctly is obtained below:
From the given information, let the random variable X be the number of correct answer follows Binomial distribution with there are 10 questions are randomly selected. That is, and the probability of guessing correct is p=14 or 0.25.
The probability mass function of X is,
P(X=x)=(nx)(p)x(1p)nx
The required probability is,
P(X=8)=[(108)(0.25)8(10.25)108]
=[(108)(0.25)8(0.75)2]
=[45×0.0000153×0.5625]
=0.00038
0.0004
Thus, the probability that R guesses exactly 8 of the questions correctly is 0.0004.
Step 2
The probability that he guesses 8 or more questions correctly is obtained below:
The required probability is,
P(X8)=P(X=8)+P(X=9)+P(X=10)
=[(108)(0.25)8(10.25)108+(109)(0.25)9(10.25)109+(1010)(0.25)10(10.25)1010]
=[(108)(0.25)8(0.75)2+(109)(0.25)9(0.75)1+(1010)(0.25)10(0.75)0]
=[45×0.0000153×0.5625+10×0.0000038×0.751×0.0000×0.75]
=0.00038+0.000028+0.0000
=0.0004
Thus, the probability that he guesses 8 or more questions correctly is 0.0004.

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