There are 12 marbles in a bag, and the marbles are either yellow or green. Two marbles will be randomly picked from the bag, without replacing the first one picked. The probability that both marbles will be yellow is 5/33

. How many YELLOW marbles are in the bag?
A)
4
B)
5
C)
6
D)
7

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

Y = # of yellow marbles

Y/12 * (Y-1)/(12-1) = 5/33

Solve for Y.

To find the number of yellow marbles in the bag, we can set up an equation based on the given probability.

Let's assume there are x yellow marbles in the bag. Since there are 12 marbles in total, we can determine the number of green marbles as (12 - x).

Now, the probability of picking a yellow marble on the first draw is x/12, since there are x yellow marbles out of 12 in total.
After the first yellow marble is removed, there would be (x-1) yellow marbles left in the bag, out of a total of (11) marbles.
So, the probability of picking another yellow marble on the second draw (without replacement) would be (x-1)/11.

Multiplying these probabilities together, we get the probability that both marbles will be yellow: (x/12) * ((x-1)/11) = 5/33.

Simplifying this equation:
33x(x-1) = 60(x-1).

Expanding and rearranging the equation:
33x^2 - 33x = 60x - 60.

Bringing all terms to one side:
33x^2 - 93x + 60 = 0.

Simplifying further, we get a quadratic equation:
11x^2 - 31x + 20 = 0.

Factoring this quadratic equation:
(11x - 4)(x - 5) = 0.

From this equation, we get two possible values for x: 4 and 5.

However, since we are looking for the number of yellow marbles, the answer cannot be 5, as there would be no yellow marble left for the second draw. Therefore, the only possible solution is x = 4.

Therefore, the number of YELLOW marbles in the bag is 4.

So, the correct answer is A) 4.

To find the number of yellow marbles in the bag, we need to set up an equation using the given probability.

Let's assume there are x yellow marbles in the bag. Since there are a total of 12 marbles in the bag, the number of green marbles would be (12 - x).

When we pick the first marble, the probability of picking a yellow marble would be (x/12) since there are x yellow marbles out of a total of 12 marbles.

After picking the first marble, we don't replace it, so there would be one less marble in the bag. Therefore, for the second pick, the probability of picking a yellow marble would be ((x - 1)/(12 - 1)).

According to the given information, the probability that both marbles will be yellow is 5/33.

So, using the multiplication rule of probability, we can set up the equation:
(x/12) * ((x - 1)/(12 - 1)) = 5/33

Simplifying the equation:
(x/12) * ((x - 1)/11) = 5/33

Multiplying both sides of the equation by 12 * 11 * 33 to eliminate the fractions:
33 * x * (x - 1) = 12 * 11 * 5

Expanding and simplifying further:
33x^2 - 33x = 660

Rearranging the equation:
33x^2 - 33x - 660 = 0

Now we can solve this quadratic equation to find the value of x.

Using quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
where a = 33, b = -33, and c = -660

Calculating:
x = (-(-33) ± √((-33)^2 - 4 * 33 * (-660))) / (2 * 33)
x = (33 ± √(1089 + 87120)) / 66
x = (33 ± √88209) / 66

Using a calculator to find the square root of 88209:
x ≈ (33 ± 297) / 66

Considering both positive and negative solutions separately:
x1 ≈ (33 + 297) / 66 ≈ 330 / 66 ≈ 5
x2 ≈ (33 - 297) / 66 ≈ -264 / 66 ≈ -4

Since we are dealing with the number of marbles, the solution -4 is not possible. Therefore, the number of yellow marbles in the bag is approximately 5.

Therefore, the answer is option B) 5.