A student is getting ready to take an important oral examination and i

Serotoninl7

Serotoninl7

Answered question

2021-11-13

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 each other, with probability .8, whereas if he has an off day, this probability will be reduced to .4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student feels 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 & Explanation

Geraldine Flores

Geraldine Flores

Beginner2021-11-14Added 21 answers

Define random variable X that marks the number of examine that the pass the student. Also define Y as the random variable that marks whether the student has "on" day or "off" day. We are given that
P(Y=1)=13, P(Y=0)=23
Suppose that are n examiners, where n is odd natural number. Let's find the distribution of X. Given the information about Y, we know that each examiner will pass student with probabilities 0.8 if Y=1 and 0.4 if Y=0 independently from each other. Hence, given Y, X has binomial distribution with parameters n and appropriate probability of succes. More precisely
P(X=k)=P(X=kY=1)P(Y=1)+P(X=kY=0)P(Y=0)
=13(nk)0.8k0.2nk+23(nk)0.6k0.4nk
Suppose not that n=3. The probability that the student will pass test is P(X2)=P(X=2)+P(X=3). We have that
P(X2)=13(32)0.830.22+23(53)0.630.42+13(54)0.840.2+23(54)0.640.4+130.85+230.650.769
So, we see that the student has higher chances to pass if there are 5 examiners.
Result: He should choose 5 examiners.

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