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  4. In an experimental performance of a single throw of a pair of unbiased normal dice, let three events E 1, E 2 and E 3 are defined as follows: E 1: getting prime numbered face on each dice. E 2: getting the same number on each dice. E 3: getting total on two dice equal to 4 . Which of the following is/are TRUE?

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Q. In an experimental performance of a single throw of a pair of unbiased normal dice, let three events $E _1, E _2$ and $E _3$ are defined as follows:
$E _1$ : getting prime numbered face on each dice.
$E _2$ : getting the same number on each dice.
$E _3$ : getting total on two dice equal to 4 .
Which of the following is/are TRUE?

Probability - Part 2

Solution:

$E _1=\{(2,2),(2,3),(3,2),(2,5),(5,2),(3,5),(5,3),(3,3),(5,5)\} $
$E_2=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\} $
$E_3=\{(1,3),(3,1),(2,2)\}$
Now $P\left(E_1\right)=\frac{9}{36} ; P\left(E_2\right)=\frac{6}{36} ; P\left(E_3\right)=\frac{3}{36}$
Clearly $P \left( E _1\right), P \left( E _2\right), P \left( E _3\right)$ are in A.P. $\Rightarrow( A )$ is correct.
$P \left( E _1 \cap E _2\right)=\frac{3}{36}=\frac{1}{12} \neq P \left( E _1\right) P \left( E _2\right) \Rightarrow$ (B) is incorrect.
Now $P\left(E_3 / E_1\right)=\frac{P\left(E_3 \cap E_1\right)}{P\left(E_1\right)}=\frac{1}{9} \Rightarrow(C)$ is incorrect.
Also $ P \left( E _1+ E _2\right)+ P \left( E _2- E _3\right)=\left[ P \left( E _1\right)+ P \left( E _2\right)- P \left( E _1 \cap E _2\right)\right]+\left[ P \left( E _2\right)- P \left( E _2 \cap E _3\right)\right.$
$=\frac{9}{36}+\frac{12}{36}-\frac{3}{36}-\frac{1}{36}=\frac{17}{36} \Rightarrow$
(D) is correct.