Skip to main content

Independent events; regulators from B and C

Short answer

Independent: P(A∩B)=P(A)P(B); mutually exclusive: P(A∩B)=0. i) P(meets spec) = 0.6×0.95 + 0.3×0.80 + 0.1×0.65 = 0.875. ii) P(B or C|meets spec) = (0.24+0.065)/0.875 = 0.3486.

Define independent and mutually exclusive events with an example. An assembly plant receives its voltage regulators from these three different suppliers, 60% from supplier A, 30% from supplier B and 10% from supplier C. It is also known that 95% of voltage regulators from A, 80% of these from B and 65% these from C perform according to specifications. What is the probability that: i) Any one voltage regulator received by the plant will perform according to specifications. ii) A voltage regulator that perform according to specification came from B and C.

Independent events

Events A and B are independent if the occurrence of one does not affect the probability of the other: P(A ∩ B) = P(A) P(B). Example: getting a head on a coin and a 6 on a die.

Mutually exclusive events

Events that cannot occur together: P(A ∩ B) = 0. Example: the same die showing 2 and 5 in one throw. Two mutually exclusive events (with non-zero probabilities) are never independent.

Voltage regulators

Let A, B, C be the suppliers and S the event "meets specification".

P(A) = 0.6   P(S|A) = 0.95   P(A ∩ S) = 0.570
P(B) = 0.3   P(S|B) = 0.80   P(B ∩ S) = 0.240
P(C) = 0.1   P(S|C) = 0.65   P(C ∩ S) = 0.065

i) Probability that a regulator meets specification

P(S) = 0.570 + 0.240 + 0.065 = 0.875

ii) Probability that a regulator that meets specification came from B or C A regulator comes from only one supplier, so "B and C" together is impossible (probability 0). The question is read as "from B or from C":

P(B or C | S) = (0.240 + 0.065) / 0.875 = 0.305 / 0.875 = 0.3486

This equals 1 − P(A|S) = 1 − 0.570/0.875 = 1 − 0.6514.

Answer: (i) 0.875, (ii) 0.3486.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 1 questions →