Sure, let's break down the problem step-by-step to calculate the probability that Mick picks two balls of different colors.
1. Total Number of Balls:
There are a total of 5 red, 4 blue, and 3 green balls in the bag. Adding these together gives a total of:
[tex]\[
5 + 4 + 3 = 12 \text{ balls}
\][/tex]
2. Total Possible Pairs of 2 Balls:
We need to calculate the number of ways to pick 2 balls from these 12 balls. This can be found using combinations (denoted as [tex]\( C(n, k) \)[/tex], where [tex]\( n \)[/tex] is the total number and [tex]\( k \)[/tex] is the number of items to choose). The number of ways to choose 2 out of 12 is:
[tex]\[
C(12, 2) = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66
\][/tex]
So, there are 66 possible pairs of balls.
3. Calculating Pairs with Different Colors:
We need to count the number of ways to pick two balls of different colors. There are three scenarios for this:
- Red and Blue: We choose 1 red ball from 5 and 1 blue ball from 4. The number of ways to do this is:
[tex]\[
5 \times 4 = 20
\][/tex]
- Red and Green: We choose 1 red ball from 5 and 1 green ball from 3. The number of ways to do this is:
[tex]\[
5 \times 3 = 15
\][/tex]
- Blue and Green: We choose 1 blue ball from 4 and 1 green ball from 3. The number of ways to do this is:
[tex]\[
4 \times 3 = 12
\][/tex]
Adding these together gives the total number of pairs with different colors:
[tex]\[
20 + 15 + 12 = 47
\][/tex]
4. Probability Calculation:
The probability that Mick picks two balls of different colors is the ratio of the number of favorable pairs (different colors) to the total number of pairs:
[tex]\[
\text{Probability} = \frac{\text{Number of pairs with different colors}}{\text{Total number of pairs}} = \frac{47}{66} \approx 0.7121
\][/tex]
Therefore, the probability that Mick picks two balls of different colors is approximately [tex]\( 0.712 \)[/tex] or about [tex]\( 71.21\% \)[/tex].
