So I was playing this game with my family, its called farkle. I don't really need to get into the rules of the game but the gist is you need to roll a 1 or a 5 with a die to score. I was thinking about this and wondering what the chances are of rolling a 1 or a 5 with x amount of dice. I thought, well the probability of rolling a 1 or 5 on one die is 2/6, but then with 2 dice it is not 2/6. I don't know exactly why it isn't 2/6, it just feels like something isn't right because of the fact that you can roll a 1 or 5 on one die and not the other, and I wanted to know how this works. Can anyone please help me understand the mathematical relation between these probabilities?
Farkle Probability of rolling 1 or 5 on at least one die with x number of dice
diceprobability
Best Answer
Each die has a $\frac 23$ chance not to show a $1$ or $5$. If you have $x$ dice, the chance that none of them show $1$ or $5$ is $\left(\frac 23\right)^x$ by the multiplication principle. The chance that you get at least one die showing $1$ or $5$ is then $1-\left(\frac 23\right)^x$