Question

A test consists of 10 true/false questions. To pass the test a student must answer at least 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test?

Answer 1

37.70% probability that the student will pass the test

Explanation:

For each question, there are only two possible outcomes. Either the student guesses it correctly, or he does not. The probability of a student guessing a question correctly 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.

10 true/false questions.

10 questions, so
`n = 10`

True/false questions, 2 options, one of which is correct. So
`p = (1)/(2) = 0.5`

If a student guesses on each question, what is the probability that the student will pass the test?

`P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)`

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

`P(X = 6) = C_(10,6).(0.5)^(6).(0.5)^(4) = 0.2051`

`P(X = 7) = C_(10,7).(0.5)^(7).(0.5)^(3) = 0.1172`

`P(X = 8) = C_(10,8).(0.5)^(8).(0.5)^(2) = 0.0439`

`P(X = 9) = C_(10,9).(0.5)^(9).(0.5)^(1) = 0.0098`

`P(X = 10) = C_(10,10).(0.5)^(10).(0.5)^(0) = 0.0010`

`P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.2051 + 0.1172 + 0.0439 + 0.0098 + 0.0010 = 0.3770`

37.70% probability that the student will pass the test