# Probability that a student passed the probability and statistics exam

The theoretical part of the exam was passed by 40% from the students, of which 80% passed the written part. On the other hand, 20% of the students which did not pass the theoretical exam, passed the written part of the exam.

a) The practical part of the exam (the third and final one) can be taken by student who passed either the written, or theoretical parts of the exam. If we know that all of the students who passed all of the previous parts, and only half of the students who passed only one of the previous parts took the practical part, what is the probability that a randomly chosen student out of all of the students, took the practical part?

b) What is the probability that a randomly chosen student that took the practical part, passed both previous exam parts?

What I have so far:

A - the event which represents the students who passed the theoretical part

B - the event which represents the students who passed the written part

P(A) = 0.4

P(B|A) = 0.8

P(A') = 0.6

P(B|A') = 0.2

a) To find the probability that a randomly chosen student took the practical part, if we know all the students that passed both parts, and half of all of the students that passed at least one previous part, took the practical part, we have to find the probability that the students passed both parts, and the probability that the students passed at least one part.

Let S denote the event in which a randomly chosen student took the practical part.

Finding the probability that the students passed both parts is as follows:

P(AB) = P(A) $$*$$ P(B|A)= 0.4 $$*$$ 0.8 = 0.32

Finding the probability that the students passed the theoretical part of the exam, but failed the written part:

P(A'B) = P(A') $$*$$ P(B|A') = 0.6 $$*$$ 0.2 = 0.12

The part where I am stick is I do not how to calculate the probability that the students passed the theoretical part, but failed the written part.

Can someone please check if my previous work is correct and any useful hints are welcome. Thanks in advance.

## 1 Answer

In these cases it is very important to desrcribe the phenomena with a table:

Taken as an example 100 students, this is the table of the frequencies of who passed (1) not passed (0) the Theoretical (T) or Written (W) part It i s very easy now to respond to any question:

(a) the probablity that a randomly chosen student took the practical part is

$$\frac{32+\frac{8+12}{2}}{100}=42\%$$

(b) Only 42 students took the practical part. Among them only 32 passed both T and W parts, thus the requested probability is

$$\frac{32}{42}=\frac{16}{21}\approx 76.19\%$$

• Is there a way to calculate the event P(AB') without the table? – ptushev Oct 23 '20 at 10:58