What is the probability of drawing a white ball from a bag containing 3 black balls and 4 white balls?

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A bag contains numbered balls: four white and five black. Three balls are drawn at random from the bag. What is the probability that all the balls are of the same color.

The probabilities that we have $(1,2,3,4)$ white balls in the bag are $$\left({1\over8},{3\over8},{3\over8},{1\over8}\right)\ .$$ Given that we have $(1,2,3,4)$ white balls in the bag the probabilities that we draw two white balls are $$\left(0,{1\over6}, {1\over2},1\right)\ .$$ The overall probability $p(2w)$ to draw two white balls then is given by $$p(2w)={1\over8}\cdot0+{3\over8}\cdot{1\over6}+{3\over8}\cdot{1\over2}+{1\over8}\cdot1={3\over8}\ .$$ Finally the probability that all three balls are white computes to $$p(3w)={1\over8}\cdot0+{3\over8}\cdot0+{3\over8}\cdot{1\over4}+{1\over8}\cdot1={7\over32}\ .$$ It follows that the conditional probability $p(3w|2w)$ is given by $$p(3w|2w)={p(3w\wedge 2w)\over p(2w)}={p(3w)\over p(2w)}={7\over12}=0.58333\ldots\ .$$

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