Say I roll a 20-sided dice 4 times. The probability of the first roll hitting 1 is $1/20$ (5%). However, what is the chance of one of the 4 dice coming up with a 1 (and 2 dice coming up with a 1, and 3 dice coming up with a 1)?
I know the probability of rolling 4 ones in a row would be $1/20^4$ (.0000625%) but not sure how to figure the rest of it out.
The probability of rolling just one 1 is the probability of rolling a 1 with one die times the probability of rolling a non-1 three times. $\frac{1}{20}\times \frac{19}{20}\times \frac{19}{20}\times \frac{19}{20}$
But that's just the probability of the first die to be a one. So we have to multiply that by 4 to get the total probability of rolling one 1. $(\frac{1}{20})^1\times(\frac{19}{20})^3\times4$
The probability of rolling two 1's is similarly done. $(\frac{1}{20})^2(\frac{19}{20})^2\times(number\space ways\space the\space 1s\space can\space be\space arranged)=(\frac{1}{20})^2(\frac{19}{20})^2\times{4\choose 2}$
Finally rolling three 1's is done the same way as rolling one 1. $(\frac{1}{20})^3\times(\frac{19}{20})\times4$