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Hello everyone.
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In this question we have three boxes, box one, box two and box three, each containing black and white balls.
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Now, the black ball is denoted by the letter capital b and the white ball is denoted by the letter capital w.
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So, box one contains six white and three black balls.
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Box 2 contains two white and three black balls and box 3 contains one white and one black ball.
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Now we draw a ball from box 1 and place it in box 2.
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Then we we draw a ball from box 2 and place it in box 3.
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Finally we draw a ball from box 3.
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So, in part a of the question, we need to to find the probability that the last ball drawn from box 3 is white and in part b of the question we need to find the probability that the last ball drawn is black now initially in box 1 we have 6 white and 3 black balls and we are drawing a ball from box 1 and placing it in box 2 so we can draw either a white ball with probability 6 divided by 9 and place it in box 2 or we can draw a black ball with probability 3 divided by 9 and place it in box 2.
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Therefore we can write in the first draw that is we are drawing a ball from box 1 and placing it in box 2.
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If we draw a white ball from box 1 and place it in box 2, so this event, let us denote it by number 1 and in that case there will be three white and three black balls in box 2.
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So the probability of three white and three black balls in box 2, this will be equal to 6 divided by 9 and event 2 is when we are drawing a black ball from box 1 to box 2 so in that case there will be a total of 2 white and 4 black balls in box 2 so the probability of this event happening is equal to 3 divided by 9 now in the second draw we are actually drawing a ball from box two and placing it in box three note that before that we had already drawn a ball from box one and placed it in box two so accordingly we will have four such possibilities or four such events in the second draw so let us denote that so in the second draw we are drawing from box to a ball and we are placing the ball in box 3 so let us say event 1 .1 we are drawing a white ball from box 1 to box 2 and then we are drawing another white ball from box 2 to box 3 so in box 3 now there will be 2 white and 1 black ball so, the probability of occurrence of this event will be equal to 3 divided by 6.
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And let us denote event 1 .2 as we draw a white ball from box 1 to box 2 and then we draw a black ball from box 2 to box 3.
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So, now in box 3, there will be one white and two black balls and the probability of occurrence of this event will be again 3 divided by 6.
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Now in event 2 .1, we are drawing a black ball from box 1 to box 2 and we are drawing a white ball from box 2 to box 3.
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Therefore, we have two white and one black ball in box 3 and the probability of the occurrence of this event will be 2 divided by 6.
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Now in event 2 .2, we are drawing a black ball from box 1 to box 2 and also we are drawing another black ball from box 2 to box 3.
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So we have 1 white and 2 black ball in box 3 and the probability of occurrence of this event will be 4 divided by 6.
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Now we analyze each and every of this chain of moves, that is from 1 to 1 .1, 1 .2, and from 2 to 2 .1 or 2 .2.
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So this chain of moves are from box 1 to box 2 and from box 2 to box 3 with their corresponding probabilities, and from there we need to get what we will have at the end of the each chain in box 3.
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So let us consider the move of 1 to 1 .1.
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So this chain of event will result in two white and one black ball in box 3 with probability.
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Since this chain of events all the events are independent of each other we need to multiply the corresponding probability.
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So, in this case it will be 6 divided by 9 multiplied by 3 divided by 6, that is 1 divided by 3.
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So in the chain of events from 1 to 1 .2, this will lead to 1 white and 2 black balls in box 3 with probability 6 divided by 9 multiplied by 3 divided by 6, that is equal to.
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To 1 divided by 3.
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Similarly, the chain of events from 2 to 2 .1, this will lead to 2 white and one black ball in box 3 with probability 3 divided by 9 multiplied by 2 divided by 6, that is equal to 1 divided by 9.
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And the chain of events 2 to 2 .2 .2 will lead to 1 white and 2 black balls in box 3 with probability 3 divided by 9 multiplied by 4 divided by 6 that is equal to 2 divided by 9...
