A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball?If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find x.

Options

A. `x = 3`
B. `x = 5`
C. `x = 6`
D. `x = 7`

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the variables
Let \( x \) be the number of black balls in the box. The total number of balls in the box is given as 12.

### Step 2: Calculate the probability of drawing a black ball
The probability \( P(B) \) of drawing a black ball can be calculated using the formula:
\[ P(B) = \frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{x}{12} \]

### Step 3: Add more black balls to the box
According to the problem, 6 more black balls are added to the box. Therefore, the new number of black balls becomes:
\[ \text{New number of black balls} = x + 6 \]
The total number of balls in the box now becomes:
\[ \text{New total number of balls} = 12 + 6 = 18 \]

### Step 4: Calculate the new probability of drawing a black ball
The new probability \( P(NB) \) of drawing a black ball after adding the black balls is:
\[ P(NB) = \frac{\text{New number of black balls}}{\text{New total number of balls}} = \frac{x + 6}{18} \]

### Step 5: Set up the equation based on the problem statement
The problem states that the new probability is double the original probability:
\[ P(NB) = 2 \cdot P(B) \]
Substituting the probabilities we calculated:
\[ \frac{x + 6}{18} = 2 \cdot \frac{x}{12} \]

### Step 6: Simplify the equation
First, simplify the right side:
\[ 2 \cdot \frac{x}{12} = \frac{2x}{12} = \frac{x}{6} \]
Now we have:
\[ \frac{x + 6}{18} = \frac{x}{6} \]

### Step 7: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 6(x + 6) = 18x \]

### Step 8: Expand and simplify the equation
Expanding the left side:
\[ 6x + 36 = 18x \]
Now, rearranging the equation:
\[ 36 = 18x - 6x \]
\[ 36 = 12x \]

### Step 9: Solve for \( x \)
Dividing both sides by 12:
\[ x = \frac{36}{12} = 3 \]

### Conclusion
The value of \( x \) is 3. Therefore, there are 3 black balls in the box.

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