Fair coin toss as example of conditional expection

Question

I have the following problem:
A fair coin is tossed. If it shows head(tail), two(six) dice are rolled. Let S be the sum of the numbers displayed by the dice. What is the expectation of S?

So the result is $E[S] = P('head')E[S|'head'] + P('tail')E[S|'tail'] = 14$
My question is how can I calculate $E[S|'head']$ and $E[S|'tail']$?

Thank you

Answer 1

Let $D_i$ be the result for throwing a single die. These will be independent and identically distributed, and for all $i$, $\mathbb E(D_i)$ is easy to find by applying the definition of expectation.

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$\mathbb E(S\mid\textsf{head})$ is the expected sum of two dice.$$\mathbb E(S\mid\textsf{head}) = \mathbb E(D_1+D_2)$$

$\mathbb E(S\mid \textsf{tail})$ is the expected sum of six dice.$$\mathbb E(S\mid\textsf{head}) = \mathbb E(D_1+D_2+D_3+D_4+D_5+D_6)$$

Both are easy to find by applying the Linearity of Expectation.

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Note: Also, since we expect to throw four dice, $\mathsf E(S)$ is the expected sum of four dice.