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Theorem: If the events $B_1,B_2,...,B_k$ constitue a partition of the sample $S$ such that $P(B_i) \neq 0$ for $i=1,2,...,k$, then for any event $\boldsymbol{R}$ of $\boldsymbol{S}$ $$P(R)=\sum_{i=1}^{k} P(B_i) P(R|B_i).$$ I found an example

enter image description here

but I can not understand what is the sample space $S$ in this example and how can $\boldsymbol{R}$ belong to $\boldsymbol{S}$? Can you help me to explain in more details. Thanks a lot!

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  • $\begingroup$ There is no $R$ in your equation. $\endgroup$ Sep 1, 2021 at 12:18
  • $\begingroup$ The $R$ in the linked image is the same as the $A$ in your post $\endgroup$
    – Moko19
    Sep 1, 2021 at 12:20
  • $\begingroup$ @KaviRamaMurthy $R$ is in the image. If the sample space is the ways of choosing a bag and then the marble, then a subset of this is the event that the marble is red $\endgroup$
    – Henry
    Sep 1, 2021 at 12:20
  • $\begingroup$ Oh I am sorry. I have a typos mistake when typing R (instead of A). I have eddited. $\endgroup$
    – Thach Tran
    Sep 1, 2021 at 14:51
  • $\begingroup$ It's much better to link to the original source than to take a screenshot. $\endgroup$ Sep 1, 2021 at 14:51

1 Answer 1

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Make a double entry table to represent $S$, with three rows that have all the possible choices of bags ($b_1,b_2,b_3$) and two columns representing all the possible choices of marble color ($B,R$). Then write all possible pairs you get in the entries. For example the first raw will be $(b_1,B),(b_1,R)$ and so on. $S$ is simply all these pairs you get in the double entry table, and $R$ is composed by the entries on the second column.

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