Question

Use the given value to find the following. (Enter your answers as fractions.)

P(A) = 0.3, P(B) = 0.4, P(A ? B) = 0.5

(a) P( Agiven B) =

(b) P (B given A) =

Answer 1

Answer: The required probabilities are

`(a)~P(A/B)=(1)/(2),\\\\(b)~P(B/A)=(2)/(3)`

Step-by-step explanation: We are given the following probabilities for the events A and B :

`P(A)=0.3,~~P(B)=0.4,~~\textup{and}~~P(A\cup B)=0.5.`

We are to find the values of the following probabilities :

`(a)~P(A/B),\\\\\\(b)~P(B/A).`

From the laws of probabilities, we have

`P(A\cup B)=P(A)+P(B)-P(A\cap B)\\\\\Rightarrow P(A\cap B)=P(A)+P(B)-P(A\cup B)\\\\\Rightarrow P(A\cap B)=0.3+0.4-0.5\\\\\Rightarrow P(A\cap B)=0.2`

Therefore, we get

`P(A/B)=(P(A\cap B))/(P(B))=(0.2)/(0.4)=(1)/(2),\\\\\\P(B/A)=(P(B\cap A))/(P(A))=(0.2)/(0.3)=(2)/(3)`

Thus, the required probabilities are

`(a)~P(A/B)=(1)/(2),\\\\\\(b)~P(B/A)=(2)/(3).`