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

Dear jsanders65,

Since there are 10 questions with 2 possible answers each, there are
2^10 = 1024 ways to complete the test.

Now consider the number of ways in which the student can pass the
test. He can answer all questions correctly, which is one way. Or he
can answer one of 10 questions correctly, which is 10 ways. Or he can
answer any two out of 10 questions correctly, which is 10*9/2 = 45
ways.

(If you're not convinced by that last calculation, look at it this
way. You want to choose a pair of items among 10, so you begin by
picking one item, which is a choice among 10. Then you pick another
item, which is a choice among the remaining 9. So it seems that there
are 10*9 ways to pick a pair among ten, except that half of these
selections are the mirror image of the other half. Hence the 10*9/2 =
45 possibilities.)

All in all, there are 1 + 10 + 45 = 56 ways to pass the test. Since
the student is picking answers randomly, we don't favor any particular
outcome, so his chance of passing the test is just 56/1024 = 0.0546875
or about 5.5%.

Regards,

leapinglizard

you are the best! thank you...

Thank you for the rating and the kind tip.

By the way, I made a pair of typos in my answer that do not affect the
calculations. In the last two sentences of the second paragraph, read
"incorrectly" for "correctly". The idea is that answering nine
questions correctly is the same as answering one incorrectly, and
getting eight right is the same as getting two wrong.

leapinglizard