I am struggling with one thing when it comes to dice roll. Is there any formula to count the number of outcomes of a dice roll when we are given a condition ?
For instance, what is the probability of seeing at least one 5 when throwing a pair of dice? By listing, I know that there are 11 outcomes i.e., (1,5) (2,5) (3,5) ( 4,5) (5,5) ( 6,5) (5,1) ( 5,2) (5,3) (5,4) (5,6).
But instead of listing, is there any formula that I can use to come up with 11 outcomes simply by plugging the values in that formula ?
Imagine, if I am rolling 3 dice or 4 dice. Then in such a situation, listing all the possible outcomes will be very time consuming. I know that I can use "1 minus the other event" approach but I am specifically looking for a formula ?
Any help is highly appreciated.
Best regards, Babar Khalid
If you're looking for the number of events (when you roll, say $n$ dice) where a specific number, say $5$ must occur, you can just calculate the total number of events and remove the cases where $5$ does not occur. The total number of events is given by $6^n$ and the number of events where $5$ does not occur is given by $5^n$ (each number can be 1,2,3,4 or 6).
So, for $n$ dice, the number of outcomes where 5 (or any particular number) occurs is given by $6^n-5^n$. For $n=2$, this gives $6^2-5^2=11$.