There are 9 jars each with unique labels.
Someone has come and removed all the labels and mixed them up.
If you put the labels back on the jars (without knowing the contents), what is the expected number of labels which would match the contents?
n = the labelling event number, 0 based
C(n) = Probability of labelling n being correct
F(n) = Probability of correct jar for labelling n free
C(n) = F(n) * (1 / (9-n))
F(0) = 1
F(n) = F(n-1) * (8-n)/(9-n),
via "diagonal" cancelling:
F(n) = 9-n/9
C(n) = (9-n)/9 * (1 / (9-n)) = 1/9.
So for all the labellings, the probability of being correct is 1/9. Because this is independently true for all cases, the labelling is "isomorphic" to 9 bernoulli trials. applying binomial theorem, we get that expected correct labellings is 1.
(the purely symbolic presence of '9' above means this generalizes to any number as fl suggested)
diagonal cancelling
