Question

The probability of event A is x, and the probaility of event B is y.

If the two events are independent, which condition must be true?
P(A|B) = x

Answer 1

The condition P(A|B) = x is true.

Explanation:

Given : The probability of event A is x, and the probability of event B is y.

If the two events are independent.

To find : Which condition must be true?

Solution :

P(A)=x and P(B)=y

If two events are independent

Then
`P(A\cap B)=P(A)* P(B)`

The formulas for conditional probabilities are

`P(A|B)=(P(A\cap B))/(P(B))\\ \\ P(B|A)=(P(A\cap B))/(P(A))`

If we substitute the independent condition in conditional probability formulas we get,

`P(A|B)=(P(A\cap B))/(P(B))=(P(A)* P(B))/(P(B))=P(A)\\ \\ P(B|A)=(P(A\cap B))/(P(A))=(P(A)* P(B))/(P(A))=P(B)`

Applying the given condition,

`P(A/B)=P(A)=x\\\\P(B/A)=P(B)=y`

Therefore, The condition P(A|B) = x is true.