A ball is picked up at random from a bag containing 12 balls out of which x are white. What is the probability that the ball picked up is white ? If 6 more white balls are put in the bag and a ball is picked up, the probability of picking up a white ball becomes twice the previous one. Complete the activity to find the value of x.
n(S) = 12, n(W) = x. Probability of getting a white ball
`P_(1) (W) = ........... = (x)/(12)" "....(1)`
6 White balls are put in the bag.
`therefore n(S) = ......., n(W) = ............`
Probability of getting a white a ball
`P_(2)(W) = (n(W))/(n(S)) = ..........." ".....(2)`
Now, `P_(2) (W) = 2 xx P_(1) (W)" "`....(Given)
`therefore (x + 6)/(18) = 2 xx ............`
Simplifying x = .............

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Define the initial conditions
We know that:
- Total number of balls in the bag, \( n(S) = 12 \)
- Number of white balls, \( n(W) = x \)

### Step 2: Calculate the probability of picking a white ball initially
The probability of picking a white ball, \( P_1(W) \), is given by the formula:
\[ P_1(W) = \frac{n(W)}{n(S)} = \frac{x}{12} \quad \text{(1)} \]

### Step 3: Update the conditions after adding more white balls
After adding 6 more white balls:
- The new total number of balls, \( n(S) = 12 + 6 = 18 \)
- The new number of white balls, \( n(W) = x + 6 \)

### Step 4: Calculate the new probability of picking a white ball
The new probability of picking a white ball, \( P_2(W) \), is:
\[ P_2(W) = \frac{n(W)}{n(S)} = \frac{x + 6}{18} \quad \text{(2)} \]

### Step 5: Set up the equation based on the given condition
According to the problem, the new probability is twice the previous probability:
\[ P_2(W) = 2 \times P_1(W) \]
Substituting the expressions from (1) and (2):
\[ \frac{x + 6}{18} = 2 \times \frac{x}{12} \]

### Step 6: Simplify the equation
First, simplify the right side:
\[ 2 \times \frac{x}{12} = \frac{2x}{12} = \frac{x}{6} \]
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:
\[ 6x + 36 = 18x \]
Rearranging gives:
\[ 36 = 18x - 6x \]
\[ 36 = 12x \]

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

### Final Answer
The value of \( x \) is \( 3 \).

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