A jar contains 10 blue marbles, 8 green marbles, and 4 red marbles. What is the probability that both marbles drawn are the same color, expressed in lowest terms?

To find the probability that both marbles drawn from the jar are of the same color, we begin by determining the total number of marbles and then the number of ways to select pairs of marbles of the same color.

First, calculate the total number of marbles:

• Blue: 10

• Green: 8

• Red: 4

Total marbles = 10 + 8 + 4 = 22 marbles.

Next, we calculate the number of ways to draw 2 marbles of the same color for each color:

1. For blue marbles:

The number of ways to choose 2 blue marbles from 10 is calculated using the combination formula ( \binom{n}{k} ), which gives:

[

\binom{10}{2} = \frac{10 \cdot 9}{2 \cdot 1} = 45

]

2. For green marbles:

The number of ways to choose 2 green marbles from 8:

[

\binom{8}{2} = \frac{8 \cdot 7}{2 \cdot 1} = 28