Question

There are three exam tickets on the table. Student takes one of the three. Every ticket has 3 questions, which means there are 9 questions in total. Questions are unique. Student knows 6 questions. Student passes the exam if : 1) student knows at least 2 questions from a ticket. 2) student knows exactly one question from given ticket and knows the answers to the two extra questions given from the other tickets. What is the probability that the student passes the exam? What is the probability the student had to answer extra questions if it is known that the student passed the exam?

Answer 1

The binomial distribution can be used to solve this.
The probability that a student knows the answer to a randomly selected question is 6/9.
Does that make sense?

Answer 2

It does.

Answer 3

So I need to learn about binomial distribution to be able to solve this probem?

Answer 4

Or can it be solved using Bayes formula?

Answer 5

Let A be the event that a student knows the answer to 2 or more questions on a ticket. To find P(A) we need to find the probability that the student knows the answers to exactly 2 questions on a ticket and the probability that the student knows the answers to exactly 3 questions on a ticket. Then these two values of probability are added to find P(A).\[\large P(2\ out\ of\ 3)=3C2\times (\frac{2}{3})^{2}\times(\frac{1}{3})^{1}\]
\[\large P(3\ out\ of\ 3)=(\frac{2}{3})^{3}\]
Can you now calculate P(A)?

Answer 6

Note that Bayes theorem deals with conditional probability and it does not apply to this question.

Answer 7

So the probability is P(2 out of 3) + P(3 out of 3) = 8/9?

Answer 8

In my textbook the answer to the first question should be 11/12.

Answer 9

Sorry, adding them gives 20/27. I squared the last part rather than cubed it.
But the answer is still different.

Answer 10

How many answers are given in the textbook to this question? I assumed that only two answers are required:
a) The answer to the question: "What is the probability that the student passes the exam?"
and
b) The answer to the question: " What is the probability the student had to answer extra questions if it is known that the student passed the exam?"

Answer 11

Yes, you are correct.
The answer to the first question should be 11/12.
The answer to the second question should be 12/77.

Answer 12

Thank you. So, finding P(2 or more) = 20/27 is just a step towards finding the solution to the first part of the question. Next we need to find the probability of the alternative way of passing, which is the probability that the student knows exactly one question from the given ticket and knows the answers to two extra questions given from the other tickets.