I have $7$ colors (white, black, green, orange, yellow, blue and red). I pick $3$ of them at random, and whichever color is drawn already is eliminated from the pool. What is the probability to draw each color?
I know for the first color, say red, the probability is $\frac 1 7$, correct? But once it has been eliminated, what is the probability of drawing each other color for the next two draws?
Best Answer
Let choose black as our favorite colour.
In the first round, you have a $\frac{1}{7}$ chance of picking black, in the second round, you have a $\frac{1}{6}$ chance of picking black, but only if you have not picked black already, which is a $\frac{6}{7}$ chance, so the chance of getting black in the second round is $\frac{6}{7} \times \frac{1}{6} = \frac{1}{7}$. In the last round, you have a $\frac{6}{7} \times \frac{5}{6} \times \frac{1}{5} =\frac{1}{7}$ chance of picking black, since you must pick a different colour in the first two rounds, and black in the last round.
So the chance of picking black is $\frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7}$, as is the chance of any other colour.