- 2068 Chaitra · 2+4 marks
Dependent events; defective bolt from machine B
Short answer
Independent: occurrence doesn't change probability; dependent: occurrence changes probability. P(B|defective) = (0.35×0.04) / (0.25×0.05 + 0.35×0.04 + 0.40×0.02) = 0.406.
Define dependent and independent events with examples. In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% of the total respectively. Of their output 5, 4 and 2 percent are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured from the machine B?
Independent events
Two events are independent if the occurrence of one does not change the probability of the other: P(A ∩ B) = P(A) P(B). Example: tossing two different coins.
Dependent events
Two events are dependent if the occurrence of one changes the probability of the other: P(A ∩ B) = P(A) P(B|A). Example: drawing two cards one after the other without replacement.
Bolt problem
Let A, B, C be the events that the bolt came from machine A, B, C, and D be the event that the bolt is defective.
| Machine | P(machine) | P(D given machine) | Product |
|---|---|---|---|
| A | 0.25 | 0.05 | 0.0125 |
| B | 0.35 | 0.04 | 0.0140 |
| C | 0.40 | 0.02 | 0.0080 |
| Total | 0.0345 |
P(D) = 0.0125 + 0.0140 + 0.0080 = 0.0345
P(B|D) = P(B) P(D|B) / P(D) = 0.0140 / 0.0345 = 0.4058
Answer: the probability that the defective bolt was made by machine B is about 0.406 (28/69).
✓ Answer checked by Claude Sonnet 5.5
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