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Should the sample space of an experiment include all the outcomes irrespective of their order. For example-

A box contains 1 red and 3 identical white balls. Two balls are drawn at random from the box in succession without replacement.

The sample space given in the textbook is WR,RW,WW But according to me the sample space should be RW1,RW2,RW3,W1R,W2R,W3R,W1W2, W1W3,W2W3

How should one decide what to use in which situation as the same book uses my method in many other questions. For example, another question says-

A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. All discs are similar in shape and size. A disc is drawn at random from the bag

The sample space given in the solution is- R1,R2,R3,R4,B1,B2,B3,Y1,Y2

What is the real logic to be used while describing the sample space?

Varun Singh
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1 Answers1

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The first experiment has two trials, and each two-letter experiment outcome is meant to express in sequence the two trial outcomes. Since the white balls are identical, your proposed outcomes $RW_1, RW_2$ and $RW_3$ are indistinguishable, so are collectively effectively just the single outcome $RW.$

The second experiment has one trial, and here the exercise apparently considers the four red balls to be non-identical, i.e., $R_1,R_2,R_3,R_4$ to be distinct outcomes.

ryang
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  • That's the problem! How to identify whether to treat them as distinct elements or as same? – Varun Singh Oct 03 '21 at 11:52
  • @VarunSingh The first exercise is unambiguous: it explicitly says "identical white balls"; so you are wrong and the book is correct. The second exercise, though, just says the red balls are "similar in size and shape"; this doesn't mean that they are identical; to be honest, there is no clear-cut interpretation here; best to rely on context (in other words, interrogate the entire exercise and ask: "in this scenario, are $R_1,R_2$ really distinct outcomes, or are they in fact the same outcome?"), and generally assume no more than necessary. – ryang Oct 03 '21 at 11:59
  • Thanks...i got the point – Varun Singh Oct 03 '21 at 13:22
  • @VarunSingh Orthogonally related, to give you a stronger sense of how we can set up (and the flexibility of) notation in probability modelling: this and this. (The choice of sample space is like the design of a definition: can be somewhat of an art, and depends on the desired level of detail.) Ultimately, just keep making sense of the problem and keeping track of what you're doing. – ryang Oct 03 '21 at 13:55