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I have another questions about the book of Rohatgi. I have thought that I must use the total probability rule for solve, but i am not sure

An urn contains five white and four black balls. Four balls are transferred to a second urn. A ball is then drawn from this urn, and it happens to be black. Find the probability of drawing a white ball from among the remaining three.

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  • $\begingroup$ Hint: Imagine that the black balls are labelled 1 to 4. What's the probability of drawing a white ball second if the first ball happens to be labelled 1? $\endgroup$ – Magma Jul 12 '19 at 23:58
  • $\begingroup$ Hint for the hint: What if the balls labeled 2 to 4 are red instead of black? $\endgroup$ – Magma Jul 13 '19 at 0:00
  • $\begingroup$ Shuffle a deck of cards, grab the top four, and observe that the first one is black. What's the probability that the next one is red? $\endgroup$ – Karl Jul 13 '19 at 0:20
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This is sort of a trick question. The complicated sampling arrangement leads you to think that a complicated calculation with the law of total probability is required, but it's all just obfuscation.

After choosing four balls, and then picking a ball from those four, each of the nine balls has an equal probability of being the one picked, so this is no different from choosing a ball uniformly at random from the original nine. After choosing the first ball, each of the remaining balls has an equal probability of being the second one chosen. There are $8$ balls left, and $5$ of them are white, so ....

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