5. We have a box containing 5 red balls and 10 green balls. We wl draw 2 of the

Question

5. We have a box containing 5 red balls and 10 green balls. We wl draw 2 of the balls. For each draw, all the balls in the urn are equally likely to be chosen (a) Find the probability that the bals are both the same color, assuming that we drawn without replacement. (That is, we draw the first bal, do not put it back in the box, and then draw the second ball.) (b) Find the probability that the balls are both the same color, assuming that the balls are drawn with replacement. (That is, we draw the first ball record its color, then replace it in the box, and draw the second ball.) (c) For both of 5a and 5b, decide if the second draw of a ball is independent of the first draw. Decide also for both questions if the first draw is independent of the second draw.

Explanation / Answer

a) Probability of both the balls are same colour = probability of both the balls are of red colour + probability of both the balls are of green colour = (5C2 / 15C2) + (10C2 / 15C2) = 10/105 + 45/105 = 55/105 = 0.524

b) Probability of both the balls are same colour = probability of both the balls are of red colour + probability of both the balls are of green colour = (5^2 / 15^2) + (10^2 / 15^2) = 25/225 + 100/225 = 125/225 = 0.556

c) In 5a) second draw is dependent of 1st draw, as we have to choose among the remaining balls after 1st draw. But in 5b) second draw is independent of 1st draw, as due to replacements we have the whole set to choose from.
In both cases the 1st draw is independent.

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