You have two 6-sided dice. You roll them and get a total of 10. You roll again and get a total of 9. If you get paid 1$ for each 5 that is rolled what is your expected winnings？
I guess it is the posterior probability：$$P(X_3 = 5\ or\ Y_3 = 5\Big| X_2 + Y_2 = 9\ and\ X_1 + Y_1 = 10).$$
Here $X_i\in\{1,2,3,4,5,6\}$ are i.i.d. and $Y_i\in\{1,2,3,4,5,6\}$ are i.i.d. (But not sure if X, Y are i.i.d.？).
$\endgroup$ 11 Answer
$\begingroup$Assuming fair dice, I believe the expected answer would be$$E[I_{X_1=5} + I_{Y_1=5} + I_{X_2=5} + I_{Y_2=5} | X_1+Y_1=10, X_2+Y_2=9]$$which is also$$ E[I_{X_1=5} + I_{Y_1=5}| X_1+Y_1=10] + E[I_{X_1=5} + I_{Y_1=5}| X_1+Y_1=9] = 2/3 + 2/4. $$
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