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प्रश्न
Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls
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उत्तर
Let X denote the number of white balls in a sample of 3 balls drawn from a bag containing 4 white and 6 red balls. Then, X can take the values 0, 1, 2 and 3.
Now,
\[P\left( X = 0 \right)\]
\[ = P\left( \text{ no white ball } \right)\]
\[ = \frac{{}^6 C_3}{{}^{10} C_3}\]
\[ = \frac{20}{120}\]
\[ = \frac{1}{6}\]
\[P\left( X = 1 \right)\]
\[ = P\left( 1 \text{ white ball } \right)\]
\[ = \frac{{}^4 C_1 \times^6 C_2}{{}^{10} C_3}\]
\[ = \frac{60}{120}\]
\[ = \frac{1}{2}\]
\[P\left( X = 2 \right)\]
\[ = P\left( 2 \text{ white balls } \right)\]
\[ = \frac{{}^4 C_2 \times^6 C_1}{{}^{10} C_3}\]
\[ = \frac{36}{120}\]
\[ = \frac{3}{10}\]
\[P\left( X = 3 \right)\]
\[ = P\left( 3 \text{ white balls } \right)\]
\[ = \frac{{}^4 C_3}{{}^{10} C_3}\]
\[ = \frac{4}{120}\]
\[ = \frac{1}{30}\]
Thus, the probability distribution of X is given by
| X | P(X) |
| 0 |
\[\frac{1}{6}\]
|
| 1 |
\[\frac{1}{2}\]
|
| 2 |
\[\frac{3}{10}\]
|
| 3 |
\[\frac{1}{30}\]
|
