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 stimulus repetition in a randomly drawn sequence of length n. We consider two probabilities, below, concerning repetition of (i) a single stimulus and (ii) 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 x 89 x 88) / 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)!] )