Question

You and your spouse each take two gummy vitamins every day. You share a single bottle of 60 vitamins, 30 of one flavor and 30 of another. You each prefer a different flavor, but it seems childish to fish out two of each type (but not to take gummy vitamins). So you just take the first four that fall out and then divide them up according to your preferences. For example, if there are two of each flavor, you and your spouse get the vitamins you prefer, but if three of your preferred flavor come out, you get two of the ones you like and your spouse gets one of each. Of course, you start a new bottle every 15 days. On average, over a 15 day period, how many of the vitamins you take are the flavor you prefer?

Answer 1

For convinience we color code the two flavor vitamins as red R and blue B

Let’s start with the first day with 60 vitamins bottle. The vitamins will come out according to the hypergeometric distribution: It describes the number of successes (the preferred R vitamin) in a random draw from a finite population (total 60 vitamins) without replacement. According to this distribution, the probability of drawing k vitamins you prefer in the first four vitamins drawn on the first day is given by:
P(k prefered vitamins) = [(30 C k) x (30 C (4-k))] / (60 C 4)
P(k=4) = 0.056
P(k=3) = 0.25
P(k=2) = 0.39
P(k=1) = 0.25
P(k=0) = 0.056
So, Expected number of preferred vitamins eaten = (0.056)(2) + (0.25)(2) + (0.39)(2) + (0.25)(1) + (0.056)(0) = 1.64 preferred vitamins each day.
So over a 15 day period, total number of preferred vitamins that you eat =
= 1.64 x 15 = 24.6 out of 30 preferred vitamins.
24.6 is roughly 82 % of 30.
Hence, over a 15 day period 82% of the vitamins you take are of the falvour you prefer.
Although the combination drawn out of the bottle on one day will affect the combination drawn out the next day, but the initial symmetry will balance thse affects in long run.