Among the examinees in an examination 30%, 35% and 45% failed in Statistics, in Mathematics, and in at least one of the subjects respectively. An examinee is selected at random. Find the probabilities that

a. He failed in Mathematics only
b. He passed in Statistics, if it is known that he has failed in Mathematics

good

a. Well, if 35% failed in Mathematics and 45% failed in at least one subject, that means 45% - 35% = 10% failed in Statistics only. Therefore, the probability that the examinee failed in Mathematics only is 10%.

b. Now, if it is known that the person failed in Mathematics, we need to find the probability of passing in Statistics. Since 45% failed in at least one subject and 35% failed in Mathematics, that means 45% - 35% = 10% failed in Statistics only. So, the probability of passing in Statistics, given that the person failed in Mathematics, is 10%.

To find the probabilities, we first need to understand the given information. Let's denote the events as follows:

A: Failing in Statistics
B: Failing in Mathematics
C: Failing in at least one subject

We are given the following probabilities:

P(A) = 30%
P(B) = 35%
P(C) = 45%

Now, let's find the probabilities step by step:

a. To find the probability that he failed in Mathematics only, we need to find P(B and not A). We can use the formula:

P(B and not A) = P(B) - P(A and B)

P(A and B) represents the probability of failing in both Statistics and Mathematics. Since this information is not given, we need to calculate it using the information about failing in at least one subject.

P(A and B) = P(C) - P(not A and not B)
P(not A and not B) represents the probability of passing in both Statistics and Mathematics.

P(not A and not B) = 1 - P(C)

Now we can substitute the values into the formula:

P(B and not A) = P(B) - (P(C) - P(not A and not B))

P(B and not A) = 35% - (45% - (1 - 30%)(1 - 35%))

Calculate the percentages and simplify the expression to find the probability.

b. To find the probability that he passed in Statistics, given that he failed in Mathematics, we need to find P(A | B), which represents the probability of passing in the Statistics subject, given that he failed in Mathematics. We can use the formula:

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

We have already calculated P(A and B) in the previous step. Now, plug the values into the formula to find the probability.

P(S and M) = P(S) + P(M) - P(S or M)

= .3 + .35 - .45
= .2

You could now make a Venn Diagram with two overlapping circles called S and M
Enter .2 into the overlap ( the P(S and M) )
since S must be .3 and .2 is already accounted for, the part of ONLY S would be .1
in the same way the part of M only would be .15