MHB Probability of Drawing Same-Colored Balls from a Box

[rymatson406]

A box contains 2 red, 4 white, and 4 green two balls are drawn in succession without replacement. What is the probability that both balls are the same color?

 

[mathmari]

A box contains 2 red, 4 white, and 4 green two balls are drawn in succession without replacement. What is the probability that both balls are the same color?

The probability is the sum of the probabilities:

• 1. ball red and 2. ball red
• 1. ball white and 2. ball white
• 1. ball green and 2. ball green

The probability "1. ball red and 2. ball red":

The probability to pick a red ball from the box which contains $10$ balls, where $2$ of them are red, is $\displaystyle{\frac{2}{10}}$.
Now there are $9$ balls left in the box and $1$ of them is red.
So the probability for the second ball to be red is $\displaystyle{\frac{1}{9}}$.

Therefore the probability "1. ball red and 2. ball red" is equal to $$\frac{2}{10} \cdot \frac{1}{9}$$Do the same for the two other cases, and you get:
$$\frac{2}{10} \cdot \frac{1}{9}+\frac{4}{10} \cdot \frac{3}{9}+\frac{4}{10} \cdot \frac{3}{9}$$