James R. answered • 09/22/19

Certified NYS / NYC Math Teacher

Solving this problem mathematically would probably need stochastic calculus, given the large number of combinations of conditional probabilities applied at random.

So, I am approaching this with the application of a series of postulates:

(1) The distribution of colors is equal. Only two colors, in equal quantity, with no preference in selection

(2) Each "draw" of 3 beans, when there are at least 3 beans available, produces one of only 4 outcomes: (a) GGG, (b) RRR, (c) GGR or (d) RRG,

(3) Given (2), and the fact that both (a) and (b) return no beans, and both (c) or (d) return 1 bean, then the likelihood of any given color being returned or not returned is equal - no preference to Red or Green.

(4) Given (1), (2) and (3) I postulate that any prediction requiring a specific color in the result will be harder (and therefore less likely) than a prediction that includes both colors with equal likelihood

(5) Given (4), I propose that the stated possible final events 2- 5 are highly unlikely, as they require the appearance of specific colors in the result

(6) Likewise, as in (5) the confidence of predicting that exactly no beans remain after a large number of random conditional events would necessarily be low, hence I postulate that the stated possible **event #1 is the least likely of all.**

(7) Finally, I would have most confidence in predicting a final event that has an equal distribution of colors, **hence #6 would be the most likely **- given that we know that fewer than 3 beans remain.

Well, here I am, out on limb. Any comments?