A cube has each of its 6 faces wieh random number chosen independently and uniformly at random from 1~6.
What is the expected number of pairs of adjacent faces with numbers differ by 1?
And also roll the cube, let T denote the sum of the numbers on all the faces of the die, X be the score X(upward-facing number) What is E(X|T)?
I don't actually see a way expect of counting the number of pairs from 1 to 12 or sth like that. Any Ideas?
On
That is really two questions. Hints:
For the first, consider each edge: what is the probability an edge has its two adjacent faces with the desired relationship? How many edges are there?
For the second, you can use a symmetry/exchangeability argument: given $T$, each face has the same expected value, and you know what these faces add up to.
Expectation is linear. Consider, for each of the 12 pairs of adjacent sides, the variable $X_n$ which is $1$ if pair $n$ are given numbers that differ by one and $0$ if they don't.
We have $X=\sum X_n$, and therefore $E(X)=\sum E(X_n)$. The right-hand side is not too difficult to calculate, as the twelve expectations are all equal. Even in the second problem, conditioned on the value of the top facing side, this is the way to go, where four of the $E(X_n\mid T)$ are distinct from the rest.