Question

A box contains a certain number of balls out of which 80% are white, 15% are blue and 5% are red. All the balls of the same color are indistinguishable. Among all the white balls, 𝛼% are marked defective, among all the blue balls, 6% are marked defective and among all the red balls, 9% are marked defective. A ball is chosen at random from the box. If the conditional probability that the chosen ball is white, given that it is defective, is 0.4, then 𝛼 equals ________

Answer 1

Correct Answer: 1.125

Let's define variables based on the problem statement. Let the total number of balls be \( N \). The number of each type of ball is: White balls = 0.8\( N \), Blue balls = 0.15\( N \), Red balls = 0.05\( N \). Now, calculate the defective balls:

• White defective balls = 0.8\( N \times \frac{\alpha}{100} \) 
• Blue defective balls = 0.15\( N \times 0.06 \)
• Red defective balls = 0.05\( N \times 0.09 \)

The conditional probability that a ball is white given it is defective is given as 0.4. Use the formula for conditional probability:

\(P(\text{White} | \text{Defective}) = \frac{P(\text{White and Defective})}{P(\text{Defective})}\)

Using probabilities:

\(\frac{0.8 \alpha / 100}{0.8 \alpha / 100 + 0.15 \times 0.06 + 0.05 \times 0.09} = 0.4\)

After simplifying:

\(\frac{0.8 \alpha / 100}{0.8 \alpha / 100 + 0.009 + 0.0045} = 0.4\)

Multiply both sides by the denominator:

\(0.8 \alpha / 100 = 0.4 \times (0.8 \alpha / 100 + 0.0135)\)

Distribute 0.4:

\(0.8 \alpha / 100 = 0.32 \alpha / 100 + 0.0054\)

Rearrange to find \( \alpha \):

\(0.8 \alpha / 100 - 0.32 \alpha / 100 = 0.0054\)

Simplify:

\(0.48 \alpha / 100 = 0.0054\)

Multiply by 100:

\(0.48 \alpha = 0.54\)

Divide by 0.48:

\(\alpha = \frac{0.54}{0.48} = 1.125\)

Verify the solution falls within the given range: 1.125 is within [1.125, 1.125]. Therefore, \( \alpha \) equals 1.125, satisfying the problem requirements.