Determine the probability that this student is a) very well prepared, b) poorly prepared.

Question

In a group of 10 students taking the exam, there are 3 who have prepared very well, 4 well, 2 moderately well and one poorly.

well, 4 well, 2 moderately and 1 poorly prepared. There are a total of 20 exam questions. One

student knows how to answer all 20 questions, one well prepared student knows how to answer

16 questions well, a moderately well prepared student answers 10 and a poorly prepared student is able to answer

answer 5 questions. A randomly selected student is able to answer all three questions asked.

questions that are asked.

Answer 1

There's probably an easier way to do this, but...

Find each combined probability: that someone is ___ prepared AND answers all three questions correctly. These are dependent, of course. Questions are selected without replacement (so out of 20, then 19, then 18).

Very Well: 3/10 * 1 * 1 * 1 = 0.3

Well: 4/10 * 16/20 * 15/19 * 14/18 = 0.196

Mod Well: 2/10 etc. = 0.021

Poor: 1/10 etc. = 0.00088

Sum of those probabilities is 0.51788 (chance that anyone answered all three questions correctly)

Now we find the conditional probabilities:

Very Well IF all three correct: 0.3 / 0.51788 = 57.9%

Poor IF all three correct: 0.1 / 0.51788 = 0.17%