Question

The Venn diagram shows the results of two events resulting from rolling a number cube.

Find P(A ∩ B).
Find P(A).
Find P(A | B).
Find P(B).

Answer 1

A)
`P(A\cap B)=(1)/(3)`

B)
`P(A)=(1)/(3)`

C)
`P(A\mid B)=(1)/(2)`

D)
`P(B)=(2)/(3)`

Explanation:

As shown in Venn diagram :

We can see that there are two outcomes that comes in the intersection part of A and B i.e. 1 and 2.

Total number of outcomes =6

So,
`P(A\cap B)=(2)/(6)=(1)/(3)`

Next, As we can see that there are only two outcomes in the part of A i.e.

`P(A)=(3)/(6)=(1)/(3)`

Now, we have to find
`P(A\mid B)`

AS we know the formula for "Conditional Probability":

`PA\mid B)=(P(A\cap B))/(P(B))\\\\PA\mid B)=((2)/(6))/((4)/(6))\\\\PA\mid B)=(2)/(4)=(1)/(2)`

Now, lat but not the least we will find P(B), as there are 4 outcomes we can see in the section of B i.e. 3, 4, 5, 6.

`P(B)=(4)/(6)=(2)/(3)`

Hence,

A)
`P(A\cap B)=(1)/(3)`

B)
`P(A)=(1)/(3)`

C)
`P(A\mid B)=(1)/(2)`

D)
`P(B)=(2)/(3)`