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QJEEM24Ma6894: JEE Main 2024 Mathematics - Class 12 JEEM24Ma6894 Solution: A bag contains 8 balls, whose colours are either white or…
A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is :
Multiple Choice Options
A
2 5 \frac{2}{5} 5 2
B
2 7 \frac{2}{7} 7 2
C
1 7 \frac{1}{7} 7 1
D
1 5 \frac{1}{5} 5 1
Step-by-Step Verified Solution
a m p ; P ( 4 W 4 B / 2 W 2 B ) = a m p ; P ( 4 W 4 B ) × P ( 2 W 2 B / 4 W 4 B ) P ( 2 W 6 B ) × P ( 2 W 2 B / 2 W 6 B ) + P ( 3 W 5 B ) × P ( 2 W 2 B / 3 W 5 B ) a m p ; + … … … … + P ( 6 W 2 B ) × P ( 2 W 2 B / 6 W 2 B ) \begin{aligned} & \mathrm{P}(4 \mathrm{~W} 4 \mathrm{~B} / 2 \mathrm{~W} 2 \mathrm{~B})= \\\\ & \frac{P(4 W 4 B) \times P(2 W 2 B / 4 W 4 B)}{P(2 W 6 B) \times P(2 W 2 B / 2 W 6 B)+P(3 W 5 B) \times P(2 W 2 B / 3 W 5 B)} \\ & +\ldots \ldots \ldots \ldots+P(6 W 2 B) \times P(2 W 2 B / 6 W 2 B)\end{aligned} am p ; P ( 4 W 4 B /2 W 2 B ) = am p ; P ( 2 W 6 B ) × P ( 2 W 2 B /2 W 6 B ) + P ( 3 W 5 B ) × P ( 2 W 2 B /3 W 5 B ) P ( 4 W 4 B ) × P ( 2 W 2 B /4 W 4 B ) am p ; + ………… + P ( 6 W 2 B ) × P ( 2 W 2 B /6 W 2 B ) a m p ; = 1 5 × 4 C 2 × 4 C 2 8 C 4 1 5 × 2 C 2 × 6 C 2 8 C 4 + 1 5 × 3 C 2 × 5 C 2 8 C 4 + … + 1 5 × 6 C 2 × 2 C 2 8 C 4 a m p ; = 2 7 \begin{aligned} & =\frac{\frac{1}{5} \times \frac{{ }^4 \mathrm{C}_2 \times{ }^4 \mathrm{C}_2}{{ }^8 \mathrm{C}_4}}{\frac{1}{5} \times \frac{{ }^2 \mathrm{C}_2 \times{ }^6 \mathrm{C}_2}{{ }^8 \mathrm{C}_4}+\frac{1}{5} \times \frac{{ }^3 \mathrm{C}_2 \times{ }^5 \mathrm{C}_2}{{ }^8 \mathrm{C}_4}+\ldots+\frac{1}{5} \times \frac{{ }^6 \mathrm{C}_2 \times{ }^2 \mathrm{C}_2}{{ }^8 \mathrm{C}_4}} \\\\ & =\frac{2}{7}\end{aligned} am p ; = 5 1 × 8 C 4 2 C 2 × 6 C 2 + 5 1 × 8 C 4 3 C 2 × 5 C 2 + … + 5 1 × 8 C 4 6 C 2 × 2 C 2 5 1 × 8 C 4 4 C 2 × 4 C 2 am p ; = 7 2
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