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a. The answer is 13.
As the women selects the balls at random without looking at them, there is a possibility that the first 10 balls which she selects are all red. Then to have atleast 3 blue balls she has to pick three balls from the rest of 10 balls. As all of them are blue only 3 picks are needed for that. Hence 10+3=13 balls she need to pick to be sure of having atleast 3 blue balls.
b. The answer is 5.
As there are 2 colours of balls present in the bowl, Red and Blue. Consider there is a Red bucket and a Blue bucket outside the bowl and after picking each ball from the bowl, each ball must go to either of the buckets. Hence applying pigeonhole principle to be sure of having atleast 3 balls of same colour
she need to pick 2+2+1=5 balls from the bowl.