As part of my recent interest in university-level statistics, I'm trying to figure out what the probability distribution is for the following scenario, without having to resort to counting the number of outcomes:
Suppose I roll two dice, but rather than adding them together I use the higher of the two as the "result" of the event.
This is not a homework question, I'm just trying to learn some "advanced" statistics techniques out of personal interest (for example, I've recently learned that you can find the probability distribution for the sum of two independent events by convolving the individual distributions). So I'd like to know if there's a "better" way to do this than enumerating all the possible outcomes and counting those that fit the criterion. I suppose the criteria for "better" would be:
- Less computationally expensive (i.e. faster if it were to be implemented on a computer)
- More interesting from a mathematical analysis standpoint.
Update to question
How would I generalize the explanation in dsaxton's answer, namely:
$P(\max(X_1, X_2) = k) = 2 P(X_1 = k \cap X_2 \leq k) - P(X_1 = k \cap X_2 = k)$
to the maximum of three or more dice?
In the case of the max of three dice, it looks like the cases are:
- $x1 = k$ and $x2 \leq k$ and $x3 \leq k$
- $x2 = k$ and $x1 \leq k$ and $x3 \leq k$
- $x3 = k$ and $x1 \leq k$ and $x2 \leq k$
which suggests that the probability would be
$P(\max(x1, x2, x3) = k) = P(x1 = k \cap x2 \leq k \cap x3 \leq k) + P(x2 = k \cap x1 \leq k \cap x3 \leq k) + P(x3 = k \cap x1 \leq k \cap x2 \leq k) - P(x1 = k \cap x2 = k \cap x3 = k)$
(though maybe this is still multi-counting?)
Is there some way to use combinatorics or some other technique to express the expanded formula for the max of $n$ dice?