# A student enrolls in a statistics class. There is a probability of 0.6 that they will be assigned...

## Question:

A student enrolls in a statistics class. There is a probability of 0.6 that they will be assigned to Professor Jones's class, and a probability of 0.4 that they will be assigned to Professor Smith's class. Historically, the probability of getting a B or higher in Professor Jones's class was 0.7, while in Professor Smith's class it was 0.9.

What is the probability that the student will get a B or better?

a. 0.78

b. 0.80

c. 0.36

d. 0.22

Given that a student achieved less than a B, what is the probability that the student had Professor Smith as an instructor?

a. 0.5385

b. 0.4615

c. 0.8182

d. 0.1818

## Bayes' Theorem & Theorem of Total Probability:

For the theorem of total probability, let {eq}A {/eq} denote an event and {eq}B_1, B_2,.... {/eq} denote a sequence of events that are mutually exclusive and the union of which makes the whole sample space. Further, let P(.) denote the probability of an event.

Then:

{eq}P(A) = \sum\limits_{i=1}^n P(A \cap B_i) \\ P(A) = \sum\limits_{i=1}^n P(A| B_i)P(B_i). {/eq}

where {eq}P(A|B_i) {/eq} denotes the probability of {eq}A{/eq} given that {eq}B_i {/eq} has already occurred.

For Bayes' theorem, let {eq}A,B {/eq} be two events. Then, Bayes' theorem is as follows:

{eq}P(A|B) = \dfrac{P(B|A)P(A)}{P(B)}. {/eq}

Let {eq}J {/eq} and {eq}S {/eq} denote the probability that a student will be assigned to Professor Jones and Professor Smith respectively.

Let {eq}B {/eq} denote the event of getting a B or higher.

We are given that:

{eq}P(J) = 0.6\\ P(S) = 0.4\\ P(B|J) = 0.7\\ P(B|S) = 0.9 {/eq}

We are required to find the probability that the student will get a B or better, that is, {eq}P(B). {/eq}

Using the theorem of total probability, we have:

{eq}P(B) = P(B|J)P(J) + P(B|S)P(S) \\ P(B) = (0.7 \times 0.6)+(0.9 \times 0.4) \\ P(B) = 0.42+0.36 \\ P(B) =0.78. {/eq}

So a is the correct option.

We have to find the probability that a student had Professor Smith as instructor, given that he achieved less than a B, that is, {eq}P(S|B') {/eq}.

Using Bayes' theorem, we have:

{eq}P(S|B') = \dfrac{P(B'|S)P(S)}{P(B')}\\ \ \\ P(S|B') = \dfrac{(1-0.9)\times 0.4}{1-0.78} \\ \ \\ P(S|B') = \dfrac{0.04}{0.22}\\ \ \\ P(S|B') = 0.1818 {/eq}

Therefore, d is the correct option.