Question

An unbiased die is thrown twice. Let the even $A$ be ${}^{\prime }od{d}^{\prime }$ number on the first throw and $B$ the event ${}^{\prime }odd$ number on the second throw'. Check the independence of the events $A$ and $B$.

Answer 1

Solution:-

Given that an unbiased die is thrown twice.

$\therefore$ Total no. of outcomes $=36$

Also given that the even $A$ be an odd number on the first throw and $B$ be an odd number on the second throw.

Therefore,

$P\left(A\right)=\frac{18}{36}=\frac{1}{2}$

$P\left(B\right)=\frac{18}{36}=\frac{1}{2}$

$P(A\cap B)=P\left(\text{odd number on both throw}\right)=\frac{9}{36}=\frac{1}{4}$

As we know that two events $A$ and $B$ are independent if $P(A\cap B)=P\left(A\right).P\left(B\right)$

Now,

$P\left(A\right).P\left(B\right)=\frac{1}{2}.\frac{1}{2}=\frac{1}{4}$

$\because P(A\cap B)=P\left(A\right).P\left(B\right)$

Hence the events $A$ and $B$ are independent.