Question

Two dice are tossed. If (m,n) denotes a typical sample point, find whether the following two events A and B are independent$A=\{(m,n):m+n=11\},B=\{(m,n):n\ne 5\}.$ If A and B are independent then enter 1, else enter 0.

Answer 1

Here $A$ consists of two elements $(6,5)$ and $(5,6)$ only whereas $B$ consists of $30$ elements of all the elements of sample space except the elements $(1,5),(2,5),(3,5),(4,5),(5,5),(6,5)$

Hence $A\cap B$ consists of a single element $(5,6)$ only.

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

$P\left(B\right)=\frac{30}{36}=\frac{5}{6}$ and $P(A\cap B)=\frac{1}{36}.$

Thus here $P\left(A\right)P\left(B\right)=\frac{1}{18}\times \frac{5}{6}=\frac{5}{108}.$

giving $P(A\cap B)\ne P\left(A\right)P\left(B\right).$

Hence $A$ and $B$ are dependent events.