Question

A six sided die is rolled six times. What is the probability that each side appears exactly once?

Answer 1

The probability is approximately 1.54%.

Explanation:

On the first roll, there are no restrictions. The die is allowed to be any of the 6 equally likely values. Thus the probability of not duplicating any numbers so far after roll 1 is `6/6`, or `100%.`

For each subsequent roll, the number of "successful" rolls decreases by 1. For instance, if our first roll was a `[3]`, then the second roll needs to be anything but `[3]`, meaning there are 5 "successful" outcomes (out of the 6 possible) for roll 2. So, since each roll is independent of the previous rolls, we multiply their "success" probabilities together. The probability of rolling no repeats after two rolls is `6/6 xx 5/6=5/6,` which is about `83.3%.`

Continuing this pattern, the third roll will have 4 "successful" outcomes out of 6, so we get

`Pr("3 unique rolls") = 6/6 xx5/6 xx 4/6=55.6%`

and then

`Pr("4 unique rolls") = 6/6 xx5/6 xx 4/6 xx3/6=27.8%`

`Pr("5 unique rolls") = 6/6 xx5/6 xx 4/6 xx3/6xx2/6=9.26%`

and finally

`Pr("6 unique rolls") = (6 xx 5 xx 4 xx 3 xx 2 xx 1)/(6^6)=1.54%.`