Four balls are simultaneously picked at random from a jar containing 2 red balls, 2 green balls, and 6 yellow balls.

What is the probability that exactly two of the selected balls will be red?

Using combination symbols,

prob = C(2,2)/C(10,2) = 1/45

or

prob = (2/10)(1/9) = 2/90 = 1/45

it just says how many red balls are there

To find the probability that exactly two of the selected balls will be red, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

1. Calculate the total number of possible outcomes:
The total number of balls in the jar is 2 red + 2 green + 6 yellow = 10 balls.
We need to choose 4 balls, so the total number of possible outcomes is given by the combination formula:
C(10, 4) = 10! / (4! * (10-4)!) = 210.

2. Calculate the number of favorable outcomes:
To have exactly two red balls, we must choose 2 red balls out of the 2 available and 2 non-red balls out of the remaining 8 balls.
The number of favorable outcomes is given by the combination formula as well:
C(2, 2) * C(8, 2) = (2! / (2! * (2-2)!) ) * (8! / (2! * (8-2)!)) = 1 * (8! / 2! * 6!) = 8 * 7 / 2 = 28

3. Calculate the probability:
Finally, divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 28 / 210 ≈ 0.1333

Therefore, the probability that exactly two of the selected balls will be red is approximately 0.1333 or 13.33%.

To find the probability of exactly two of the selected balls being red, we need to calculate the total number of favorable outcomes and divide it by the total number of possible outcomes.

First, let's determine the total number of possible outcomes. We are picking four balls from a jar with a total of 2 red balls, 2 green balls, and 6 yellow balls. Thus, there are a total of 10 balls to choose from.

We can calculate the total possible outcomes using the combination formula, denoted as C(n, r), which represents the number of ways to choose r items from a set of n distinct items. In this case, we want to choose 4 balls from a total of 10, so the total number of possible outcomes is:

C(10, 4) = 10! / (4! * (10-4)!) = 210

Now, let's determine the total number of favorable outcomes. We want exactly two of the selected balls to be red. This can be achieved in two ways: choosing 2 red balls and 2 other balls (green or yellow), or choosing 1 red ball, 1 other red ball, and 2 other balls (green or yellow).

The number of ways to choose 2 red balls from the available 2 is C(2, 2) = 1.
The number of ways to choose 2 other balls (green or yellow) from the available 8 is C(8, 2) = 28.
Alternatively, the number of ways to choose 1 red ball from the available 2 is C(2, 1) = 2.
The number of ways to choose 1 other red ball from the remaining 1 is C(1, 1) = 1.
The number of ways to choose 2 other balls (green or yellow) from the remaining 8 is C(8, 2) = 28.

Therefore, the total number of favorable outcomes is:

1 * 28 + 2 * 1 * 28 = 168

Finally, we can calculate the probability by dividing the total number of favorable outcomes by the total number of possible outcomes:

Probability = Favorable outcomes / Total outcomes = 168 / 210 = 4 / 5 = 0.8

So, the probability that exactly two of the selected balls will be red is 0.8 or 80%.