Question

Refer to question 1 above. If the die were fair, determine whether or not the events A and B are independent.

Answer 1

Referring to the above solution, we haveA = {(1, 1) (2, 2) (3, 3) (4, 4) (5, 5), (6, 6)}

$\Rightarrow n\left(A\right)=6andn\left(S\right)=6^2=36$ [Where, S is sample space]

$\therefore P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{6}{36}=\frac{1}{6}$

and $B=\left\{\right(4,6),(6,4),(5,5),(6,5),(5,6),(6,6\left)\right\}$

$\Rightarrow n\left(B\right)=6andn\left(S\right)=6^2=36\therefore P\left(B\right)=\frac{n\left(B\right)}{n\left(S\right)}=\frac{6}{36}=\frac{1}{6}$
Also, $A\cap B=\left\{\right(5,5),(6,6\left)\right\}$
$\Rightarrow n(A\cap B)=2andn\left(S\right)=36\therefore P(A\cap B)=\frac{2}{36}=\frac{1}{18}$
Also, $P\left(A\right).P\left(B\right)=\frac{1}{36}$
Thus, $P(A\cap B)\ne P\left(A\right).P\left(B\right)$ $[\because \frac{1}{18}\ne \frac{1}{36}]$
So, we can say that both A and B are not independent events.