What is the probability of observing exactly one 3 when rolling a fair die five times?
Same as for rolling 5 dice at the same time, namely 5 x 1/6 x
(5/6)4 This is the second last term in the expansion of (a + b)5
where a is the probability of a die falling at 3 (ie 1/6) and b is
the probability of it not falling at 3 (ie 5/6) and the various
terms of the expansion correspond to the various combinations : a5
+ 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. Which of course adds up to 1
when you allow all possibilities, not just one 3.