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So here is a basic probability problem and its answer which I extracted from a MIT tutorial:

Suppose that in front of you are two bowls, labeled A and B. Each bowl contains five marbles. Bowl A has 1 blue and 4 yellow marbles. Bowl B has 3 blue and 2 yellow marbles. Now choose a bowl at random and draw a marble uniformly at random from it. Based on your existing knowledge of probability, how likely is it that you pick a blue marble? How about a yellow marble? Out of the 10 marbles you could choose from, 4 are blue. So the probability of choosing a blue marble is 4 out of 10.

My question is - why would you consider picking from either bowl the same as picking from a single bowl containing all 10 marbles. If the marbles are in two bowls, even if you are free to pick from either bowl, the moment you choose a bowl, your sample space is now restricted to the number of marbles in that bowl and not all the marbles - isn't it? Could someone please help me understand this? Thanks in advance.

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Yes, it is the same as long as the count of marbles are the same in both bowls.

There are two independent actions: choosing one of two bowls and choosing one of five marbles. The probability of choosing bowl x is $\frac12$ and the probability of choosing marble y in a given bowl is $\frac15$. So the probability of choosing marble y from bowl x is $\frac1{10}$ and this is the same for all marbles.

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  • $\begingroup$ So its 1/2 the times 4/5 plus half the times 2/5 - gives the same answer 6/10 for the probability of picking a yellow marble. But its not the same as considering the sample space to be all 10 marbles. Makes sense. Thanks a lot. $\endgroup$ – dillee May 22 at 12:37

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