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Q.
A bag contains $5$ white, $4$ black and $2$ red balls. Balls are drawn one by one without replacement. The probability that the $5^{t h}$ ball is a red ball, is
NTA AbhyasNTA Abhyas 2022
Solution:
If the $5^{t h}$ ball is a red ball $\Rightarrow $ either the first four ball are of other colour and $5^{t h}$ ball is red or the first four ball consist of $1$ red ball, $3$ other ball and $5^{t h}$ ball is red ball
Hence, the required probability $= \frac{\_{}^{9}C _{4}^{}}{\_{}^{11}C _{4}^{}} \times \frac{2}{7} + \frac{\_{}^{9}C _{3}^{} \times \_{}^{2}C _{1}^{}}{\_{}^{11}C _{4}^{}} \times \frac{1}{7}$
$=\frac{9 \times 8 \times 7 \times 6}{11 \times 10 \times 9 \times 8}\times \frac{2}{7}+\frac{9 \times 8 \times 7 \times 2 \times 4}{11 \times 10 \times 9 \times 8}\times \frac{1}{7}$
$=\frac{6}{55}+\frac{4}{55}=\frac{10}{55}=\frac{2}{11}$