A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let $\mathrm{X}$ be the number of white balls, among the drawn balls. If $\sigma^{2}$ is the variance of $\mathrm{X}$, then $100 \sigma^{2}$ is equal to ________.
Answer (integer)
57
Solution
<p>$X =$ Number of white ball drawn</p>
<p>$P(X = 0) = {{{}^6{C_3}} \over {{}^{10}{C_3}}} = {1 \over 6}$</p>
<p>$P(X = 1) = {{{}^6{C_2} \times {}^4{C_1}} \over {{}^{10}{C_3}}} = {1 \over 2},$</p>
<p>$P(X = 2) = {{{}^6{C_1} \times {}^4{C_2}} \over {{}^{10}{C_3}}} = {3 \over {10}}$</p>
<p>and $P(X = 3) = {{{}^6{C_0} \times {}^4{C_3}} \over {{}^{10}{C_3}}} = {1 \over {30}}$</p>
<p>Variance $= {\sigma ^2} = \sum {{P_i}X_i^2 - {{\left( {\sum {{P_i}{X_i}} } \right)}^2}}$</p>
<p>$${\sigma ^2} = {1 \over 2} + {{12} \over {10}} + {3 \over {10}} - {\left( {{1 \over 2} + {6 \over {10}} + {1 \over {10}}} \right)^2}$$</p>
<p>$= {{56} \over {100}}$</p>
<p>$100{\sigma ^2} = 56.$</p>
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
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