Let B be the number of coins that are blue on one side.
Let R be the number of coins that are red on both sides.
It would be impossible to solve this problem without some sort of assumption of the probability distribution of B or R. We will assume that B is a random integer between 0 and 600, evenly distributed.
The first question is obvious. The number of blue coins showing most likely makes up about half of B. Therefore, the best estimate is B = 400 and R = 200.
The second question is not so obvious. We must consider each possible value of B, and multiply the probability of that particular value of B by the probability that such a value would result in the colors we see.
probability = p(B) = 1/601 * C(B,200)/2^B * C(B,250)/2^B
C(a,b) is the combination function: C(a,b) = a!/((a-b)!b!)
The most likely value of B is the one that gives the maximum probability. One good way of finding the maximum probability is by finding the highest value of B usch that p(B)/p(B-1) is greater than 1.
p(B)/p(B-1) > 1
B^2/(B-200)/(B-250)/4 > 1 (where B >= 250)
B^2 > 4B^2 - 1800B + 200000
0 > 3B^2 - 1800B + 200000
B < ( 1800 + SQRT(1800^2 - 4 * 3 * 200000) )/6
B < 452.7525
Therefore the most likely B is 452, and R is 148. |
