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The probability of event A is x, and the probability of event B is y. If the two events are independent, which of these conditions must be true?

AP(B|A) = y
BP(A|B) = y
CP(B|A) = x
DP(A and B) = x + y

EP(A and B) = x/y

P(A)

Verified Answer :

Hagrid

The right answer for the question that is being asked and shown above is that: "d. P(A and B) = x + y." The probability of event A is x, and the probability of event B is y. If the two events are independent, the condition must be true is this d. P(A and B) = x + y

frika

If two events are independent, then

\(Pr(A\cap B)=Pr(A)\cdot Pr(B)\).

Use formulas for conditional probabilities:

\(Pr(A|B)=\dfrac{Pr(A\cap B)}{Pr(B)},\\ \\
Pr(B|A)=\dfrac{Pr(A\cap B)}{Pr(A)}\).

For independent events these formulas will be:

\(Pr(A|B)=\dfrac{Pr(A\cap B)}{Pr(B)}=\dfrac{Pr(A)\cdot Pr(B)}{Pr(B)}=Pr(A),\\ \\
Pr(B|A)=\dfrac{Pr(A\cap B)}{Pr(A)}=\dfrac{Pr(A)\cdot Pr(B)}{Pr(A)}=Pr(B)\).

Now in your case \(Pr(A)=x,\ Pr(B)=y\) and \(Pr(A|B)=x,\ Pr(B|A)=y, Pr(A\cap B)=x\cdot y\).

This shows that the only correct choice is A.

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