A jar contains x yellow marbles, x-2 white marbles, and x+2 black marbles. The probability of randomly picking 2 yellow marbles is $\frac{1}{12}$.
What is the total number of marbles in the jar?

The correct answer is 9.

Step 1: Understand the given marble counts and find the total number of marbles.

Let $x$ represent the number of yellow marbles.
The jar contains:
• Yellow marbles = $x$
• White marbles = $x-2$
• Black marbles = $x+2$

The total number of marbles in the jar is calculated by adding the quantities of all three types of marbles:

$\text{Total marbles}=x+(x-2)+(x+2)=3x$

Step 2: Express the probability of picking 2 yellow marbles.

The probability of picking the first yellow marble is the number of yellow marbles divided by the total number of marbles:

$P\left(\text{1st yellow}\right)=\frac{x}{3x}=\frac{1}{3}$

After one yellow marble is removed, there are $x-1$ yellow marbles remaining and a total of $3x-1$ marbles left in the jar.

The probability of picking the second yellow marble is:

$P\left(\text{2nd yellow}\right)=\frac{x-1}{3x-1}$

The combined probability of drawing 2 yellow marbles consecutively without replacement is:

$P\left(\text{2 yellow marbles}\right)=\frac{1}{3}\times \frac{x-1}{3x-1}$

Step 3: Set up the equation and solve for $x$.

We are given that the probability of randomly picking 2 yellow marbles is $\frac{1}{12}$:

$\frac{1}{3}\times \frac{x-1}{3x-1}=\frac{1}{12}$

Multiply both sides of the equation by 3:

$\frac{x-1}{3x-1}=\frac{3}{12}=\frac{1}{4}$

Cross-multiply to solve for $x$:

$4(x-1)=1(3x-1)$

$4x-4=3x-1$

$4x-3x=4-1$

$x=3$

Step 4: Calculate the total number of marbles.

Since the total number of marbles is $3x$:

$\text{Total marbles}=3\times 3=9$

Therefore, the total number of marbles in the jar is 9.