Question

Each student taking the probability calculus exam draws a card from 2 out of 40 different questions. In order to pass the exam, you have to answer both questions from the card or one question from the card and an additional question indicated by the examiner from another card. Max went to the exam, but knew the answer to only 33 questions. Calculate the probability that Max

(a) he passed the exam,
(b) answered both questions from a drawn sheet of card, if known to have passed the examination.
(c) answered one question on a drawn sheet of card if known not to have passed the examination.

Answer 1

Question (a)

MaX can pass the exam in two cases

1) if he answers both the questions from the cards he has picked

2) Answes one of the questions from the card he has picked and the other from the card examiner has picked

Probability that Max answers both the questions from the cards he has picked = 33/40 * 32/39

= 0.6769

The probability in first case is 33/40 * 32/38 becuase Max knows tthe answer to 33 of the questions out of 40, so in his first pick his probability will be 33/40 and second pick it will be 32/39 since 1 question is already answered

Probability that Max answers one of the questions from the card he has picked and the other from the card examiner has picked = 33/40 * 1/32

= 0.0258

The probability in second case is 33/40 * 1/32 becuase Max knows tthe answer to 33 of the questions out of 40, so in his first pick his probability will be 33/40 and in the second pick examiner has to select 1 out of the remaining 32 qusestions for Max to pass the exam

So probability Max passed the exam = 0.6769 + 0.0258 = 0.7027

Question (b)

Given that he has passed the exam, we need to find the probability that he answered both questions from a drwan sheet of cards

Here we need to use conditional probability

P (A | B) = P(AB) / P(B)

Here A is the he answered both questions from a drwan sheet of cards

B is the event that he passed the exam

P(AB) is the probability that he he answered both questions from a drwan sheet of cards and has passed the exam which is first case of Question (a)

so P(A | B) = 0.6769 / 0.7027

= 0.9633

Question (c)

Here we need to find conditional probability again

P (A | B) = P(AB) / P(B)

A is the event that he answered one question from the draw

B is the event that he did not pass the exam

P(AB) = Probability that answered one question from the draw and did not pass the exam

Probability he answered the first question from the draw and did not pass the exam = 33/40 * 1/7

becuase he answered the first question the only possibility of him faling the exam is when the examiner picks the card which is out of 7 questions that Max can not answer

So P (A | B) = 0.1176 / (1 - 0.7027)

P(not passing the exam) = 1 - P(passing the exam)

So P(A | B) = 0.1176 / 0.2973

= 0.3964