Guessing the outcome of a pair of dice using the outcome of another pair of dice

Question

I have two dice with four faces, in which the numbers ${1,2,3,4}$ are written. The total outcome, $X$, is the sum of the outcomes of each die, $X=X_1+X_2$. If a friend wants to guess the outcome of my dice, $X$, using the outcome of his own dice (which also have four faces), how can I calculate the probability that his outcome $Y$ has the same value that the outcome I got, $P(Y=x|X=x)$?

I see that $X$ and $Y$ have the same probability distribution, but I don’t see how to get this conditional probability…

Answer 1

The most reasonable hypothesis in absence of more information is that you and your friend's results are independent, which results in $$P(Y=x|X=x)=P(Y=x).$$

Answer 2

HINT

Find that probability distribution (e.g. $P(X=2)=\frac{1}{16}$, $P(X=3)=\frac{2}{16}$, etc.), and then calculate

$$\sum_{x=2}^8 P(X=Y=x)$$

where by $P(X=Y=x)$ I of course mean $P(X=x \cap Y=x)$, which works out to:

$$P(X=x \cap Y=x)= P(X=x)\cdot P(Y=x)=P(X=x)\cdot P(X=x) = P(X=x)^2$$

So, you need to calculate:

$$\sum_{x=2}^8 P(X=x)^2$$

Intuitively, this makes sense: what is the probability of getting the same outcome? It is the probability of both of you getting $2$, or both of you getting $3$, or both getting $4$ ...

Answer 3

Basic approach. Presumably, you know how to obtain the probability distribution

$$ f_X(x) = P(X = x) $$

And you have correctly recognized that your friend's dice will produce the same distribution

$$ f_Y(y) = P(Y = y) = f_X(y) $$

So then you simply have to add up the probabilities that they are equal, for every value that they might take on (from $2$ through $8$). Assuming independence, this is simply the square that your set of dice produce each value. That is, the desired probability is

$$ p = \sum_{x = 2}^8 [f_X(x)]^2 $$