Problem 19 Solution

Call the four colors 1,2,3, and 4.

After the first turn you will have a configuration like 1,1,2,3. Call this configuration 1.

From configuration 1 if you draw a 1 on the first ball you will end up with the same or similar configuration. The probability of this happening is 1/2.

From configuration 1 if you draw a 3 or 4 first and then a 1 you will have a configuration like 1,1,1,2. The probability of this happening is 1/3. Call this new pattern configuration 2.

From configuration 1 if you draw a 3 or 4 first and then draw the other non 1 you will have a configuration like 1,1,2,2. The probability of this is 1/6. Call this new pattern configuration 3.

From configuration 2 you will end up with the same pattern with probability 1/2, configuration 3 with probability 1/4, and ending the experiment with probability 1/4.

From configuration 3 you will end up with the same thing with probability 1/3, and with configuration 2 with probability 2/3.

From this information you can draw the following equations, where a is the expected number of turns from equation 1, b from equation 2, and c from equation 3:

a=(a+1)/2 + (b+1)/3 + (c+1)/6
b=(b+1)/2 + (1)/4 + (c+1)/4
c=(c+1)/3 + (2/3)*(b+1)

Finally simple matrix algebra will show a=8. Considering the first turn from the original condition to equation 1 the expected number of turns is 9.

Michael Shackleford, A.S.A.