Consider a bag of 7 blocks. Each block has a different color. The colors are Red, Orange, Yellow, Green, Blue, Indigo and Violet.
You reach into the bag of 7 blocks and pull out 2 blocks. What is the probability that at least one of the blocks you pulled out is Red, Orange or Yellow?
Best Answer
We'll take Graham's advice and think about the complements - that's a slightly easier calculation.
We want to know the probability that one of the two blocks is one of red, orange, or yellow. The complement of this event is that neither of the first two blocks is one of these three colors.
There are four blocks in the bag that are not one of the desired colors, and seven total blocks. So, the probability that the first block is not red, orange, or yellow is $\frac{4}{7}$.
If the first block is not one of our colors, then the block that was removed first was one of the four we did not want: green, blue, indigo, or violet. So, three of these remain in the bag. We removed one of the seven, so six total remain in the bag. This means that the probability that the second block is also not red, orange or yellow (given that the first one was not) is $\frac{3}{6}$.
Then, the probability that the first block was not one of the three colors and the second block was not is
$$ \frac{4}{7}*\frac{3}{6} = \frac{2}{7} . $$
We want the probability of the complement, so the probability we're looking for is
$$ 1 - \frac{2}{7} = \frac{5}{7} . $$