"Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2+\sigma^2)$ is equal to :"

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3 rotten apples are mixed with 7 good apples.

$\therefore$ Total apples = 10

Among those 10 apples 4 are chosen randomly.

${x_i}$	${p_i}$	${p_i}{x_i}$	${p_i}{({x_i})^2}$
0	${{{}^7{C_4}} \over {{}^{10}{C_4}}} = {{35} \over {210}}$	0	0
1	${{{}^3{C_1} \times {}^7{C_3}} \over {{}^{10}{C_4}}} = {{105} \over {210}}$	${{105} \over {210}}$	${{105} \over {210}}$
2	${{{}^3{C_2} \times {}^7{C_2}} \over {{}^{10}{C_4}}} = {{63} \over {210}}$	${{126} \over {210}}$	${{252} \over {210}}$
3	${{{}^3{C_3} \times {}^7{C_1}} \over {{}^{10}{C_4}}} = {7 \over {210}}$	${{21} \over {210}}$	${{63} \over {210}}$

${x_i}$ = Number of rotten apples drawn.

${p_i}$ = Probability of rotten apple.

We know,

Mean $(\mu ) = \sum {{p_i}{x_i}}$

$= 0 + {{105} \over {210}} + {{126} \over {210}} + {{21} \over {210}}$

$= {{252} \over {210}} = {6 \over 5}$

Also,

Variance $({\sigma ^2}) = \left( {\sum {{p_i}{{({x_i})}^2}} } \right) - {\mu ^2}$

$$ = {{105} \over {210}} + {{252} \over {210}} + {{63} \over {210}} - {{36} \over {25}}$$

$$ = {1 \over 2} + {{12} \over {10}} + {3 \over {10}} - {{36} \over {25}} = {{14} \over {25}}$$

$\therefore$ $10(\mu^2 + {\sigma ^2})$

$= 10\left( {({6 \over 5})^2 + {{14} \over {25}}} \right)$

$= 10\left( {{{36 + 14} \over {25}}} \right)$

$= 10 \times {{50} \over {25}} = 20$

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Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2+\sigma^2)$ is equal to :

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Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2+\sigma^2)$ is equal to :

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