Answer to Question #256616 in Statistics and Probability for Wulam Jnr

Question

Among a group of 60 engineering students, 24 take math and 29 take physics. Also 10 take both physics and statistics, 13 take both math and physics, 11 take math and statistics, and 8 take all three subjects, while 7 take none of the three. a) How many students take statistics? b) What is the probability that a student selected at random takes all three, given he takes statistics?

Answer 1

It is useful to draw the Euler circles for that situation

We can assume next things:

1). There is 13-8=5 students who take both math and physics, but not statistics(green colour)

2). There is 10-8=2 students who take both statistics and physics, but not math(grey colour)

3). There is 11-8=3 students who take both math and statistics but not physics(yellow colour)

4). There is 29-5-8-2=14 students who take only physics(pink colour)

Now we can calculate the amount of students who take only statistics next way: The amount of students taking at least one(60-7)minus the amount taking only math minus the amount taking only physics minus both physics and statistics but not math.

53-24-14-2=13 students take only statistics

So, there are 13+2+3+8=26 students take statistics

b) Let A-"student take all three", B-"student take statistics". The point is to find conditional probability

$P(A/B) = {\frac {P(A⋂B)} {P(B)}}$

$P(A⋂B) = {\frac 8 {60}}$ ;$P(B) = {\frac {26} {60}}$

So,

$P(A/B) = {\frac {P(A⋂B)} {P(B)}}={\frac 8 {26}}={\frac 4 {13}}$