a student fails to study for a 9 question true/false test. What is the probability of him getting 5 answers correct?

the answer is 63/256.

how is that the answer??

Wow, I would never have done this like this and it took me a while to figure out how they got that fractional answer. I got the decimal version (.246) by doing a binomial.

A probability is the number of ways to get the "favorable" outcome divided by the number of ways to get any possible outcome.

i.e. how many ways to get 5 of 9 of them correct divided by how many ways to do the entire test.

I actually figured out the denominator of that first. For each question, there are two possible outcomes: true or false. You do this step 9 times. This is the multiplication rule. Number of outcomes for the first step times the number of outcomes for the second step, and so on. So 2 x 2 x 2... a total of 9 times. Or 2^9. Which is 512. So there are 512 possible ways you could answer this test. (Seems incredible, but true.)

Now for the numerator.. the number of ways for the 'favorable' outcome. The favorable outcome is getting 5 of them correct. So you have a combination of ways to pick 5 out of 9. i.e. how many ways can you get 5 of the 9 answers correct?



(I only figured that out cause I had the 126 already written down from doing my binomial and then realized how it fit into this.)

So 126 ways you can pick 5 of 9. And 512 total possible ways to answer the test.

126/512 reduces to 63/256.

I would never have thought to do it this way, but it worked so I'm assuming my reasoning is correct. If someone else could confirm this... because despite it coming out correct, something's bugging me about that numerator.