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

1 Answer


The probability is approximately 1.54%.


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%.#