1. Section B ( 40 marks) a. A factory employs three machine operators to produce its brand of electric bulbs. Operator A works $$ 50 \% $$ of the time, operator B works $$ 30 \% $$ of the time and operator C works $$ 20 \% $$ of the time. Each operator is prone to produce defective bulbs. Operator A produces defective bulbs $$ 1 \% $$ of the time, operator B produces defective bulbs $$ 5 \% $$ of the time and C produces defective bulbs $$ 7 \% $$ of the time. If a defective bulb is produced, what is the probability that it was produce by i. Operator A ii. Operator B iii. Operator C b. A bag contains 3 red balls and 7 black balls. What is the probability that the first ball drawn is red and the second ball is green, given that the draws are made i. with replacement ii. without replacement

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Given:
- Operator A works 50% of the time and produces defective bulbs 1% of the time.
- Operator B works 30% of the time and produces defective bulbs 5% of the time.
- Operator C works 20% of the time and produces defective bulbs 7% of the time.

To find the probability that a defective bulb was produced by each operator, we need to calculate the probability of each operator producing a defective bulb and then sum these probabilities.

Let's denote:
- $$ P(A) $$ as the probability that a defective bulb was produced by Operator A.
- $$ P(B) $$ as the probability that a defective bulb was produced by Operator B.
- $$ P(C) $$ as the probability that a defective bulb was produced by Operator C.

The probability of a defective bulb being produced by each operator can be calculated using the formula:
$$ P(A) = \frac{0.5 \times 0.01}{0.5 \times 0.01 + 0.3 \times 0.05 + 0.2 \times 0.07} $$
$$ P(B) = \frac{0.3 \times 0.05}{0.5 \times 0.01 + 0.3 \times 0.05 + 0.2 \times 0.07} $$
$$ P(C) = \frac{0.2 \times 0.07}{0.5 \times 0.01 + 0.3 \times 0.05 + 0.2 \times 0.07} $$

Now, let's calculate these probabilities.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.5\times 0.01}{\left(0.5\times 0.01+0.3\times 0.05+0.2\times 0.07\right)}$$
- step1: Remove the parentheses:
  $$\frac{0.5\times 0.01}{0.5\times 0.01+0.3\times 0.05+0.2\times 0.07}$$
- step2: Multiply the numbers:
  $$\frac{0.5\times 0.01}{0.005+0.3\times 0.05+0.2\times 0.07}$$
- step3: Multiply the numbers:
  $$\frac{0.5\times 0.01}{0.005+0.015+0.2\times 0.07}$$
- step4: Multiply the numbers:
  $$\frac{0.5\times 0.01}{0.005+0.015+0.014}$$
- step5: Multiply the numbers:
  $$\frac{0.005}{0.005+0.015+0.014}$$
- step6: Add the numbers:
  $$\frac{0.005}{0.034}$$
- step7: Convert the expressions:
  $$\frac{\frac{1}{200}}{\frac{17}{500}}$$
- step8: Multiply by the reciprocal:
  $$\frac{1}{200}\times \frac{500}{17}$$
- step9: Reduce the numbers:
  $$\frac{1}{2}\times \frac{5}{17}$$
- step10: Multiply the fractions:
  $$\frac{5}{2\times 17}$$
- step11: Multiply:
  $$\frac{5}{34}$$
Calculate or simplify the expression $$ 0.3*0.05/(0.5*0.01+0.3*0.05+0.2*0.07) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.3\times 0.05}{\left(0.5\times 0.01+0.3\times 0.05+0.2\times 0.07\right)}$$
- step1: Remove the parentheses:
  $$\frac{0.3\times 0.05}{0.5\times 0.01+0.3\times 0.05+0.2\times 0.07}$$
- step2: Multiply the numbers:
  $$\frac{0.3\times 0.05}{0.005+0.3\times 0.05+0.2\times 0.07}$$
- step3: Multiply the numbers:
  $$\frac{0.3\times 0.05}{0.005+0.015+0.2\times 0.07}$$
- step4: Multiply the numbers:
  $$\frac{0.3\times 0.05}{0.005+0.015+0.014}$$
- step5: Multiply the numbers:
  $$\frac{0.015}{0.005+0.015+0.014}$$
- step6: Add the numbers:
  $$\frac{0.015}{0.034}$$
- step7: Convert the expressions:
  $$\frac{\frac{3}{200}}{\frac{17}{500}}$$
- step8: Multiply by the reciprocal:
  $$\frac{3}{200}\times \frac{500}{17}$$
- step9: Reduce the numbers:
  $$\frac{3}{2}\times \frac{5}{17}$$
- step10: Multiply the fractions:
  $$\frac{3\times 5}{2\times 17}$$
- step11: Multiply:
  $$\frac{15}{34}$$
Calculate or simplify the expression $$ 0.2*0.07/(0.5*0.01+0.3*0.05+0.2*0.07) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.2\times 0.07}{\left(0.5\times 0.01+0.3\times 0.05+0.2\times 0.07\right)}$$
- step1: Remove the parentheses:
  $$\frac{0.2\times 0.07}{0.5\times 0.01+0.3\times 0.05+0.2\times 0.07}$$
- step2: Multiply the numbers:
  $$\frac{0.2\times 0.07}{0.005+0.3\times 0.05+0.2\times 0.07}$$
- step3: Multiply the numbers:
  $$\frac{0.2\times 0.07}{0.005+0.015+0.2\times 0.07}$$
- step4: Multiply the numbers:
  $$\frac{0.2\times 0.07}{0.005+0.015+0.014}$$
- step5: Multiply the numbers:
  $$\frac{0.014}{0.005+0.015+0.014}$$
- step6: Add the numbers:
  $$\frac{0.014}{0.034}$$
- step7: Convert the expressions:
  $$\frac{\frac{7}{500}}{\frac{17}{500}}$$
- step8: Multiply by the reciprocal:
  $$\frac{7}{500}\times \frac{500}{17}$$
- step9: Reduce the numbers:
  $$7\times \frac{1}{17}$$
- step10: Multiply:
  $$\frac{7}{17}$$
The probabilities that a defective bulb was produced by each operator are:
- $$ P(A) \approx 0.147059 $$
- $$ P(B) \approx 0.441176 $$
- $$ P(C) \approx 0.411765 $$

Therefore, the probability that a defective bulb was produced by Operator A is approximately 14.7%, by Operator B is approximately 44.1%, and by Operator C is approximately 41.1%.
The probability that a defective bulb was produced by: - Operator A: approximately 14.7% - Operator B: approximately 44.1% - Operator C: approximately 41.1% For part b: - If the draws are with replacement, the probability of drawing a red ball first and a green ball second is 0. - If the draws are without replacement, the probability is (3/10) * (0/9) = 0. Therefore, in both cases, the probability is 0.

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