Distribution of the lowest value among $n$ throws of a die

Question

Throwing a die $N $times, the random variable $X$ is the lowest value from all of the $N$ throws. For ($1 \leq i \leq 6$) how to find $P(X=i)$?

Example: $n=5$, throws=$(6,3,4,4,5) \implies X=3$

Answer 1

For completeness, with $i \in \{0,1,2,3,4,5,6\}$

• the probability all $N$ dice values are greater than or equal to $i$ is $\left(\frac{7-i}{6}\right)^N$

• the probability all $N$ dice vales are strictly greater than $i$ is $\left(\frac{6-i}{6}\right)^N$

• the probability the lowest value of $N$ dice is exactly $i$ is $\left(\frac{7-i}{6}\right)^N - \left(\frac{6-i}{6}\right)^N$

So the probability the lowest value of $5$ dice is exactly $3$ is $\left(\frac{4}{6}\right)^5 - \left(\frac{3}{6}\right)^5 =\frac{781}{7776} \approx 0.1$. The distribution with $N=5$ dice is

Lowest  Prob
  1    0.59812
  2    0.27019
  3    0.10044
  4    0.02713
  5    0.00399
  6    0.00013