Question

Two dice are thrown together. Let A be the event ‘getting 5 on the first die’ and B be the event ‘getting 3 on the second die’. Are the events A and B independent?

• A. Yes
• B. No

Answer 1

The correct option is A Yes

There are 36 elements in the sample space.$A=\left((5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\right\}B=\left((1,3),(2,3),(3,3),(4,3),(5,3),(6,3)\right\}ThenA\cap B=(5,3)$
$P\left(A\right)=\frac{6}{36}=\frac{1}{6}P\left(B\right)=\frac{6}{36}=\frac{1}{6}P\left(A\cap B\right)=\frac{1}{36}$

Events A and B will be independent if
$P(A\cap B)=P\left(A\right)\times P\left(B\right)$

$Now,P\left(A\right)\times P\left(B\right)=\frac{1}{6}\times \frac{1}{6}=P\left(A\cap B\right)i.e.,P\left(A\cap B\right)=P\left(A\right)\times P\left(B\right)$
Hence, A and B are independent.