We pay $\$42$ so we can throw $3$ fair $6$ sided dice. We get back the product of the resulting dice values.
Is it worth playing this game?
What is the expected value of your winnings (or losings) after $10$ rounds of this game?
From how I understand it, the expected value after $10$ rounds should be $(3.5^3)*10$. Also each round is $3.5^3$ which is a little over 42, so it should be worth playing. Is this correct? How would you write it formally?
Let $Z$ be the random variable corresponding to the product of the 3 (independent, fair) dice, when you roll them. Then $Z=U_1U_2U_3$, where the $U_i$ are iid random variables, uniform on $\{1,\dots,6\}$. In one experiment, when you play, the expected gain is $$ \mathbb{E} Z-42 = \mathbb{E}[U_1U_2U_3] - 42 = \mathbb{E}[U_1]\mathbb{E}[U_2]\mathbb{E}[U_3] - 42 = \left(\mathbb{E}[U_1]\right)^3 - 42 = \left(\frac{7}{2}\right)^3 - 42 $$ where we used independence of the $U_i$. Repeating it 10 times multiplies this expected gain by 10; the expected gain is therefore $g = 10\cdot\left(\mathbb{E} Z-42\right)$ it is worth playing, thus, if $ g \geq 0. $