Question

For a loaded die the probability of outcomes are given as under.
$p\left(1\right)=p\left(2\right)=0.2,p\left(3\right)=p\left(5\right)=p\left(6\right)=0.1$ and $p\left(4\right)=0.3$ The die is thrown two times. Let A and B be the events, 'same number each time' and 'a total score is 10 or more ', respectively . Determine whether or not A and B are independent.

Answer 1

Foraloadeddie,itisgiventhat
$p\left(1\right)=p\left(2\right)=0.2$
$p\left(3\right)=p\left(6\right)=0.1$ and $p\left(4\right)=0.3$
Also,dieisthrowntwotimesHere,A=SamenumbereachtimeandB=Totalscoreis10ormore
$\therefore A=(1,1),(2,2)(3,3)(4,4)(5,5)(6,6)$

So, $P\left(A\right)=\left[P\right(1,1)+P(2,2)+P(3,3)+P(4,4)+P(5,5)+P(6,6)$
$=P\left(1\right).P\left(1\right)+P\left(2\right).p\left(2\right)+p\left(3\right).p\left(3\right)+p\left(4\right).p\left(4\right)+p\left(5\right).p\left(5\right)+p\left(6\right).P\left(6\right)$

$=0.2\times 0.2+0.2\times 0.2+0.1\times 0.1+0.3\times 0.3+0.1\times 0.1+0.1\times 0.1$
$=0.04+0.04+0.01+0.09+0.01+0.01=0.20$
And $B=\left\{\right(4,6),(6,4),(5,5),(5,6),(6,5),(6,6\left)\right\}$
$P\left(B\right)=P(4,6)+P(6,4)+P(5,5)+P(5,6)+P(6,5)+P(6,6)$

$=P\left(4\right)P\left(6\right)+P\left(6\right).\left(4\right)+P\left(5\right).\left(5\right)+P\left(5\right).\left(6\right)+P\left(6\right).\left(5\right)+P\left(6\right).\left(6\right)$
$=0.3\times 0.1+0.1\times 0.3+0.1\times 0.1+0.1\times 0.1+0.1\times 0.1+0.1\times 0.1$
$=0.03+0.03+0.01+0.01+0.01+0.01=0.10$
Also,$A\cap B=(5,5),(6,6)$
$\therefore P(A\cap B)=P(5,5),P(6,6)=P\left(5\right).P\left(5\right)+P\left(6\right).P\left(6\right)$

$=0.1\times 0.1+0.1\times 0.1=0.02$
Weknowthat,fortwoeventsAandB,if$P(A\cap B)=P\left(A\right).\left(B\right)$

,thenbothare independentevents
Here, $p(A\cap B)=0.02$ and $P\left(A\right).P\left(B\right)=0.20\times 0.10=0.02$
Hence A and B are independent events