Question

\use the Venn diagram to calculate probabilities.

Which probabilities are correct? Check all that apply.

P(A|C) = 2/3
P(C|B) = 8/27
P(A) = 31/59
P(C) = 3/7
P(B|A) = 13/27

Answer 1

Answer: The first and the third.

The two statements that are correct are the first and third options.

In the first choice, we are looking for values that are in A given that they are already a part of B. 14 of the values in B are also in A. This can be reduced to 2/3 as shown.

In the third choice, we are simply looking for the fraction of the entire chart that are in A. There are 31 values in A and 59 in the total chart. Therefore, the fraction 31/59 is correct.

Answer 2

Answer : 1 and 3 are the correct probabilities.

→According to the given Venn diagram.

Total number of elements = 59.

1)P(C)=
`(21)/(59)` and
`P(A\cap C)=(14)/(59)` then

`P(A|C)=(P(A\cap C))/(P(C))`
`=((14)/(59))/((21)/(59))=(14)/(21)=(2)/(3)`

2)P(B)=
`(27)/(59)` and
`P(C\cap B)=(11)/(59)` then

`P(C|B)=(P(C\cap B))/(P(B))`
`=((11)/(59))/((27)/(59))=(11)/(27)`
`\\eq (8)/(27)`

3) P(A) =
`(number\ of\ elements\ in\ A)/(Total\ elements)=(31)/(59)`

4) P(C) =
`(number\ of\ elements\ in\ C)/(Total\ elements)=(21)/(59)`
`\\eq (3)/(7)`

5)
`P(B|A)=(P(B\cap A))/(P(A))`
`=((13)/(59))/((31)/(59))=(13)/(31)`
`\\eq (13)/(27)`

Therefore, option 1 and 3 are correct.