China Mathematical Competition

Question

Suppose there are 8 white balls and 2 red balls in a packet. Each time one ball is drawn and replaced by a white one. Then the probability of drawing out all of the red balls just in the fourth draw is ______.

Accepted Answer

The following three cases can satisfy the condition. | | 1st draw | 2nd draw | 3rd draw | 4th draw | |--------|----------|----------|----------|----------| | Case 1 | Red | White | White | Red | | Case 2 | White | Red | White | Red | | Case 3 | White | White | Red | Red | So the probability align* P &= P(Case 1) + P(Case 2) + P(Case 3) &= 210 (910)^2 110 + 810 210 910 110 & + (810)^2 210 110 &= 0.0434. align*

	1st draw	2nd draw	3rd draw	4th draw
Case 1	Red	White	White	Red
Case 2	White	Red	White	Red
Case 3	White	White	Red	Red