Repetition probabilities
Suppose we have a sequence of length 6 (k) taken from a possible 92 (K) stimuli and wish to consider the probability of a repetition of any single stimulus in a randomly drawn sequence then we consider two probabilities below concerning repetition of a single stimulus and an entire sequence of stimuli.
The probability of any of the six of the 92 stimuli repeating in a randomly drawn sequence of length six
= 1 – [(92 x 91 x 90 x89 x88) / 92^6] = 0.99
= 1 – (number of sequences of length six which have no repetition of any of the 92 stimuli e.g. ABCDEF) / (total number of possible sequences of length 6 chosen from 92 stimuli)
so we are almost sure to get a single stimulus repeated in a randomly chosen sequence of length 6.
Another repetition which we may be interested in is the probability of an entire sequence of length 6 taken from 92 stimuli repeating in n independent draws:
This equals 1 – P(no repetition of any sequence in the n draws)
= 1 – $$ Product(i=0, n-1) [926 – i] / [92 6 ] $$
since there are 92^6 possibly distinct sequences of 92 stimuli of length 6 and once we have used one we don’t want to use it again.
which from this website
= 1 - $$( [926 !] / [(926 )n (926 - n)!] )