Consider the random experiment in which three fair dice are rolled at the same time (and also independently). Let $X$ be the random variable defined as the amount of the worths of these three dice. Let $Y_1$ be the maximum of the 3 values, let $Y_2$ be their product, and also let $Y=Y_1+Y_2$. Finally, specify $Z=E$ (the conditional expectation of $X$ given $Y$).

Find the meant value of the random variable $Z$. Go into your answer as a fraction, such as $frac32$.

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

I would certainly favor some help on wbelow to start this difficulty, please. I know $X$ can be any kind of worth between $3$ and $18$ and also $Y_2$ could be (a lot of of the values) between $1$ and $216$; i.e., $Y_2$ will never before be $7$, $11$, $13$, and so on since those numbers are prime and also are higher than $6$.

Thanks!


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

Conditional expectation has actually the complying with residential or commercial property, known as averaging: $E( E(X|Y) ) = EX$.


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Three dice having actually sides labelled $e,i,π,1,0,sqrt2$ are rolled. Find the probability of acquiring the product of the 3 results a genuine number.
What is the supposed 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, shed dollar.
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