Q. 295.0( 1 Vote )

# There are 3 red and 5 black balls in bag ‘A’; and 2 red and 3 black balls in bag ‘B’. One ball is drawn from bag ‘A’ and two from bag ‘B’. Find the probability that out of the 3 balls drawn one is red and 2 are black.

Answer :

Given:

⇒ Bag A contains 3 red balls and 5 black balls

⇒ Bag B contains 2 red balls and 3 black balls

It is told that one ball is drawn from bag A and two balls from bag B.

We need to find the probability that one ball is red and other two are black.

Let us find the Probability of drawing each colour ball from the bag.

⇒ P(B_{1}) = P(drawing black ball from bag A)

⇒

⇒

⇒

⇒ P(R_{1}) = P(drawing Red ball from bag A)

⇒

⇒

⇒

⇒ P(B_{21}) = P(drawing black ball from bag B in 1^{st} draw)

⇒

⇒

⇒

⇒ P(R_{21}) = P(drawing Red ball from bag B in 1^{st} draw)

⇒

⇒

⇒

⇒ P(B_{22}) = P(drawing black ball from bag B in 2^{nd} draw after drawing red ball)

⇒

⇒

⇒

⇒ P(R_{22}) = P(drawing Red ball from bag B in 2^{nd} draw after drawing Black ball)

⇒

⇒

⇒

⇒ P(B_{221}) = P(drawing black ball from bag B in 2^{nd} draw after drawing black ball)

⇒

⇒

⇒

We need to find the probability of drawing a red and two black balls from two bags

⇒ P(S) = P(drawing one red ball and two Black balls)

⇒ P(S) = P(drawing red ball from bag A and black balls from bag B) + P(drawing black ball from bag A and red and black balls from bag B)

Since drawing a ball is independent for each bag, the probabilities multiply each other.

⇒

⇒

⇒

⇒ .

∴ The required probability is .

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