Let two fair six-faced dice A and B be thrown simultaneously. If E₁ is event that die A show

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14.Probability

hard

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2 $ is the event that die $B$ shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true $?$

$E_1 $ and $E_3 $ are independent.

$E_1 , E_2$ and $E_3 $ are independent.

$E_1$and $E_2$ are independent.

$E_2 $ and $E_3 $ are independent.

(JEE MAIN-2016)

Solution

$E_{1} \rightarrow A$ shows up 4

$\mathrm{E}_{2} \rightarrow \mathrm{B}$ shows up 2

$E_{3} \rightarrow \operatorname{Sum}$ is odd (i.e. even $+$ odd or odd $+$ even)

$\mathrm{P}\left(\mathrm{E}_{1}\right)= \frac{6}{6.6}=\frac{1}{6}$

$\mathrm{P}\left(\mathrm{E}_{2}\right)= \frac{6}{6.6}=\frac{1}{6}$

$\mathrm{P}\left(\mathrm{E}_{3}\right)= \frac{3 \times 3 \times 2}{6.6}=\frac{1}{2} $

${\rm{P}}\left( {{{\rm{E}}_1} \cap {{\rm{E}}_2}} \right) = \frac{1}{{6.6}} = {\rm{P}}\left( {{{\rm{E}}_1}} \right) \cdot {\rm{P}}\left( {{{\rm{E}}_2}} \right)$

$ \Rightarrow \mathrm{E}_{1} \, and \, \mathrm{E}_{2} \text { are independent } $

${\rm{P}}\left( {{{\rm{E}}_1} \cap {{\rm{E}}_3}} \right) = \frac{{1.3}}{{6.6}} = {\rm{P}}\left( {{{\rm{E}}_1}} \right) \cdot {\rm{P}}\left( {{{\rm{E}}_3}} \right)$

$ \Rightarrow {{\rm{E}}_{1\,}}{\rm{and}}\,{{\rm{E}}_3}{\rm{ are independent }}$

${\rm{P}}\left( {{{\rm{E}}_2} \cap {{\rm{E}}_3}} \right) = \frac{{1.3}}{{6.6}} = \frac{1}{{12}} = {\rm{P}}\left( {{{\rm{E}}_2}} \right) \cdot {\rm{P}}\left( {{{\rm{E}}_3}} \right)$

$ \Rightarrow \mathrm{E}_{2}\,and \, \mathrm{E}_{3} \text { are independent }$

$\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2} \cap \mathrm{E}_{3}\right)=0$ ie imposible event.

Standard 11

Mathematics

If $A$ and $B$ are two events of a random experiment, $P\,(A) = 0.25$, $P\,(B) = 0.5$ and $P\,(A \cap B) = 0.15,$ then $P\,(A \cap \bar B) = $

easy

If $A$ and $B$ are any two events, then $P(\bar A \cap B) = $

normal

An unbiased die is thrown twice. Let the event $A$ be 'odd number on the first throw' and $B$ the event 'odd number on the second throw '. Check the independence of the events $A$ and $B$.

easy

If from each of the three boxes containing $3$ white and $1$ black, $2$ white and $2$ black, $1$ white and $3$ black balls, one ball is drawn at random, then the probability that $2$ white and $1$ black ball will be drawn is

hard

(IIT-1998)

Events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12}$ and $\mathrm{P}$ $($ not $ \mathrm{A}$ or not $\mathrm{B})=\frac{1}{4} .$ State whether $\mathrm{A}$ and $\mathrm{B}$ are independent?

medium