There is a box with six green marbles, nine red marbles, and seven blue marbles. Someone chooses three marbles from the box at random without looking. What is the probability that they will all be the same color?
There are 22 marbles total: 6 green, 9 red, and 7 blue. We can calculate three different probabilities for each color and then add them up.
Green: There are 6 green marbles, and we want to choose 3. The number of ways to do this is given by C(6,3) = 6!/(3!(6-3)!) = 20. So there are 20 ways to choose 3 green marbles from the 6. Now we need to find the total number of ways to choose 3 marbles from the 22. This is given by C(22,3) = 22!/(3!(22-3)!) = 22*21*20/(3*2*1) = 1540. So the probability of choosing 3 green marbles is 20/1540.
Red: There are 9 red marbles and we want to choose 3. The number of ways to do this is given by C(9,3) = 9!/(3!(9-3)!) = 84. So the probability of choosing 3 red marbles is 84/1540.
Blue: There are 7 blue marbles and we want to choose 3. The number of ways to do this is given by C(7,3) = 7!/(3!(7-3)!) = 35. So the probability of choosing 3 blue marbles is 35/1540.
Now we just need to add up the probabilities for each color: 20/1540 + 84/1540 + 35/1540 = 139/1540. So the probability of choosing 3 marbles of the same color is 139/1540 or approximately 0.09026.