Ellen takes a multiple-choice quiz that has 5 questions, with 4 answer choices for each question. What is the probability that she will get at least 2 answers correct by guessing?

Question

kinggeorge · Answer

The easiest way to do this is to find the probability that she will only get 0 or 1 answer correct, and then subtract that from 1.

kinggeorge · Answer

Do you know how to find the probability that she will get 0 correct?

anonymous · Answer

not really this stuff confuses me

kinggeorge · Answer

Well, for each question she has a $3/4$ chance of getting it wrong. To get it wrong 5 times in a row, the probability is $$\left({3 \over 4}\right)^5={243 \over 1024}$$This is the probability that she gets exactly 0 of them correct.

kinggeorge · Answer

As for the probability that she gets exactly 1 correct, the formula is a bit more weird. There are 5 questions, and she gets 1 of them wrong, so we can find the probability that she gets the first right, and rest wrong, and then multiply by 5. 

We can do that because it is equally likely that she gets the 2nd, 3rd, 4th, or 5th wrong as it is that she gets the 1st wrong.

kinggeorge · Answer

The probability that she gets the first right and the rest wrong is$${1 \over 4}\left({3 \over 4}\right)^4={81 \over 1024}$$Now we multiply by 5 to get $$5\cdot {81 \over 1024}={ 405 \over 1024}$$

kinggeorge · Answer

Now we add the probability that she got 0 to the probability that she got 1, and we get $${243 \over 1024}+{405 \over 1028}={648 \over 1024}={81 \over 128}$$If we subtract this from 1, then we get $$1-{81 \over 128}={128 \over 128}-{81 \over 128}={47 \over 28}\approx0.367$$So this is the probability that she gets at least one correct answer.

anonymous · Answer

Oh my gosh thank you so much for helping me out with this and you really explained it so good!! Thanks =)

kinggeorge · Answer

You're welcome.