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.


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.

  • $\begingroup$ 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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