Question

Question (3) [4 Marks] Note: Do not use R, do the calculations by hand.

A student is getting ready to take an important oral examination and is concerned about the possibility of having an "on" day or an “off" day. He figures that if he has an on day, then each of his examiners will pass him, independently of one another, with probability 0.8, whereas if he has an off day, this probability will be reduced to 0.4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?

Answer 1

Let the probability that the student has an on day be "x"

the probability that the student has an off day will be "2x"

x + 2x = 1, so x = 1/3

Each of the examiners on an on day will pass him with probability of 0.8 independent of each other

Each of the examiners on an off day will pass him with probability of 0.4 independent of each other

Probability of him passing the exam with 3 examiners will be

x* (0.8*0.8*0.8) + 2x* (0.4*0.4*0.4)

0.512x + 0.128x

= 0.64x

Probability of him passing the exam with 5 examiners will be

x*(0.8*0.8*0.8*0.8*0.8) + 2x* (0.4*0.4*0.4*0.4*0.4)

0.327x + 0.02x

= 0.348x

We know that x = 1/3

Probability of him passing the exam with 3 examiners will be = 0.2133

Probability of him passing the exam with 5 examiners will be = 0.1160

So he should go with 3 examiners