Answer to Question #206424 in Statistics and Probability for Weam

Question

For some NONEMPTY independent events A and B we have ‏Pr(A) = Pr(B) = Pr(AnB),then

Select one:

A. Event A is impossible and event B is certain

B. Both A and B are impossible events

C. Both A and B are certain events

D. Event A is certain and event B is impossible

Answer 1

Solution:

Since, A and B are independent events, then

$P(A \cap B)=P(A).P(B) \\=P(A).P(A)=[P(A)]^2$

But it is given that $P(A \cap B)=P(A)$

Then, $[P(A)]^2=P(A)$

$\Rightarrow [P(A)]^2-P(A)=0 \\\Rightarrow P(A)[P(A)-1]=0 \\ \Rightarrow P(A)=0 \ or \ P(A)=1$

If $P(A)=0 \ then \ P(B)=0$

And if $P(A)=1 \ then \ P(B)=1$

Thus, all 4 options are true.