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A test contains twenty true false questions. A student either knows the answer to a question, or guesses at random if he does not know the answer. On any question, the student knows the answer with probability .7. What is the probability that he gets exactly 2 of the twenty questions wrong, if his performance on different questions is independent

User Npskirk
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Answer:

2.78% probability that he gets exactly 2 of the twenty questions wrong

Explanation:

For each question, there are only two possible outcomes. Either he knows the answer, or he does not. The probability of him knowing the answer for a question is independent of other questions. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.


P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)

In which
C_(n,x) is the number of different combinations of x objects from a set of n elements, given by the following formula.


C_(n,x) = (n!)/(x!(n-x)!)

And p is the probability of X happening.

On any question, the student knows the answer with probability .7.

This means that
p = 0.7

What is the probability that he gets exactly 2 of the twenty questions wrong, if his performance on different questions is independent

2 of 20 wrong, 20-2 = 18 correctly. So this is P(X = 18).


P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)


P(X = 18) = C_(20,18).(0.7)^(18).(0.3)^(2) = 0.0278

2.78% probability that he gets exactly 2 of the twenty questions wrong

User Rico Neitzel
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