My uncle’s ritual for dressing each morning except Sunday includes a trip to the sock drawer, where he:
(1) picks out three socks at random, then
(2) wears any matching pair and returns the odd sock to the drawer or
(3) returns the three socks to the drawer if he has no matching pair and repeats steps (1) and (3) until he completes step (2).
The drawer starts with 16 socks each Monday morning (eight blue, six black, two brown) and ends up with four socks each Saturday evening.
(a) On which day of the week does he average the longest time to dress?
(b) On which day of the week is he least likely to get a pair from the first
three socks chosen?
Source: manchi tutorials
If the probability of getting a pair is p, then the expected number of attempts is 1/p. So, at first blush, I thought that (a) and (b) had the same answer, but after thinking for a few seconds I realized that it is not necessarily so.
The issue is that after Monday, there is not a single known p. Instead, p is a distribution of probabilities, which depend on what happened on earlier days. And the inverse of the expected value of the p is not the same as the expected value of the inverse of p. (a) and (b) could therefore be different days, and I am hoping (and guessing) that the number of sock pairs was selected so that (a) and (b) are in fact different days.
I'm afraid that I can't do the calculation just now, but I promise to check whoever attempts an answer.
Edited on June 28, 2013, 3:30 pm