A bag contains 3 black balls, 4 green balls and 5 yellow balls.

a) if two balls are picked at random without replacement, find the probability that Botha balls are of the same color.
b) How many black balls must be added to the bag so that the probability of drawing a black ball is 1/2.

ways of picking two balls ... 12C2 ... 66

... two black balls ... 3C2 ... 3
... two green balls ... 4C2 ... 6
... two yellow balls ... 5C2 ... 10

p(both same color) = (3 + 6 + 10) / 66

b) (3 + x) / (12 + x) = 1/2

thank you

a) Well, well, well, let's calculate the probability of picking two balls of the same color. First, we need to figure out the total number of balls in the bag: 3 black + 4 green + 5 yellow, which gives us 12 balls in total. Now, let's think of the possibilities. We have three cases: 2 black balls, 2 green balls, or 2 yellow balls. The probability of randomly picking 2 black balls is (3/12) * (2/11) because after the first ball is picked, there are only 11 balls remaining, and 2 of them are black. Following the same logic, the probability of picking 2 green balls is (4/12) * (3/11), and the probability of picking 2 yellow balls is (5/12) * (4/11). Finally, we add up these three probabilities: (3/12) * (2/11) + (4/12) * (3/11) + (5/12) * (4/11). Calculate that and you'll have the answer!

b) Ah, the magic of probability! To make the probability of drawing a black ball equal to 1/2, we need to add a specific number of black balls to the bag. Let's call that number "x". Now, to find out how many black balls we need to add, we need to set up an equation. The probability of drawing a black ball after adding "x" black balls should be (3 + x) / (12 + x) = 1/2. Cross-multiply and solve the equation to find the magical number "x". I'd tell you the answer, but where's the fun in that? Think of it as a math puzzle, my friend!

a) To find the probability that both balls are of the same color when two balls are picked at random without replacement, we need to consider the different possibilities:

1. Picking 2 black balls:
The probability of picking the first black ball is 3/12 (since there are 3 black balls out of a total of 12 balls). After picking the first black ball, there are only 2 black balls left out of 11 balls. Therefore, the probability of picking the second black ball is 2/11. So the probability of picking 2 black balls without replacement is (3/12) * (2/11) = 6/132.

2. Picking 2 green balls:
The probability of picking the first green ball is 4/12, and after picking the first green ball, there are 3 green balls left out of 11 balls. Therefore, the probability of picking the second green ball is 3/11. So the probability of picking 2 green balls without replacement is (4/12) * (3/11) = 12/132.

3. Picking 2 yellow balls:
The probability of picking the first yellow ball is 5/12, and after picking the first yellow ball, there are 4 yellow balls left out of 11 balls. Therefore, the probability of picking the second yellow ball is 4/11. So the probability of picking 2 yellow balls without replacement is (5/12) * (4/11) = 20/132.

To find the probability that both balls are of the same color, we need to sum up the probabilities of the three cases above:
Probability = (6/132) + (12/132) + (20/132) = 38/132 = 19/66.

Therefore, the probability that both balls are of the same color is 19/66.

b) Let's assume we need to add "x" black balls to the bag so that the probability of drawing a black ball is 1/2.

Initially, the bag contains 3 black balls out of a total of 12 balls. After adding "x" black balls, the total number of balls becomes 12 + x. The number of black balls in the bag would be 3 + x.

So, the probability of drawing a black ball from the bag would be (3 + x)/(12 + x).

According to the question, we need to find the value of "x" for which this probability is 1/2.

(3 + x)/(12 + x) = 1/2

Now, we can solve this equation to find the value of "x":
2(3 + x) = 12 + x
6 + 2x = 12 + x
2x - x = 12 - 6
x = 6

Therefore, we need to add 6 black balls to the bag so that the probability of drawing a black ball is 1/2.

P(black,black) = 3/12 * 2/11

do the same steps for P(green,green) and P(red,red)
then add them all up, since the only ways to get two of the same color is to get two blacks, greens, or reds.

There are 12 balls in all
There are 3 black balls
so, you need to add x black balls to make them 50% --
12+x = 2(3+x)

a) you got 3 cases: both black, both green, or both yellow

without replacement, prob of both black = (3/12)(2/11) = 1/22
without replacement, prob of both green = .....
without replacement, prob of both yellow = ...
add them up

b) number of blacks to be added ---- x
So now you have x+3 black balls out of 12+x in total
prob (black) = (x+3)/(x+12) = 1/2
solve for x