A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is [2023]

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Solution

Q.
A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is [2023]

1 $\frac{2}{7}$

2 $\frac{3}{7}$

3 $\frac{5}{7}$

4 $\frac{5}{6}$

Ans.
(3)

Total possibilities :

Case I : 2B + 4 others

Case II : 3B + 3 others

Case III : 4B + 2 others

Case IV : 5B + 1 other

Case V : 6B + 0 other

$Required probability = \frac{\frac{{}^{5}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{6}{C_{2}}}{{}^{6}{C_{2}}}}{\frac{{}^{2}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{3}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{4}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{5}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{6}{C_{2}}}{{}^{6}{C_{2}}}}$

$= \frac{10 + 15}{1 + 3 + 6 + 10 + 15} = \frac{25}{35} = \frac{5}{7}$

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