Pick 3 out of 5 days
Assume you have 3 balls, one yellow, one blue, one red. Each day, for 5 days, you pick one ball, with replacement. What is the probability that you will choose each color at least once?
I understand that there are 243 combinations (3×3×3×3×3). My thought process is to eliminate all outcomes that include: a) only 2 of the 3 colors (ie. Can't do YYYRR or YRRRR, since you didn't get blue), and b) eliminate all outcomes that include only 1 of the 3 colors (ie. Can't do RRRRR, BBBBB, or YYYYY).
For a) There are 3 ways to choose only 2 out of the 3 colors, so it should be 3 × (2/3)^5 = 96 ways
For b) There are 3 ways to choose only 1 out of the 3 colors, so it should be 3 × (1/3)^5 = 3 ways
In total there are 96 + 3 unfavorable outcomes out of 243, so there should be 144 favorable outcomes out of 243, giving me a final answer of 59.2%
In checking with AI, they did (243 - 96 + 3) / 243 instead. How come they added the 3 instead of subtracting?