Consider the random experiment in which 3 fair dice are rolled simultaneously (and also independently). Let $X$ be the random variable identified as the sum of the worths of these 3 dice. Let $Y_1$ be the maximum of the 3 worths, let $Y_2$ be their product, and also let $Y=Y_1+Y_2$. Finally, define $Z=E$ (the conditional expectation of $X$ given $Y$).

Find the meant value of the random variable $Z$. Enter your answer as a portion, such as $frac32$.

You are watching: What is the expected sum of the numbers that appear when three fair dice are rolled

I would prefer some aid on where to start this trouble, please. I recognize $X$ deserve to be any type of value in between $3$ and $18$ and $Y_2$ could be (a lot of of the values) in between $1$ and also $216$; i.e., $Y_2$ will never be $7$, $11$, $13$, and so on since those numbers are prime and are higher than $6$.




The expected worth of a die roll is 3.5. Hence, for three independent rolls, the intended worth is 10.5. $frac212$, if you"re being picky.

Conditional expectation has actually the following home, recognized as averaging: $E( E(X|Y) ) = EX$.


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Three dice having sides labelled $e,i,π,1,0,sqrt2$ are rolled. Find the probcapability of obtaining the product of the three results a actual number.
What is the intended value? Three dice are rolled. For a 1 dollar bet you win 1 dollar for each 6 that shows up (plus dollar back). No 6, lose dollar.

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