Question

A service process has three serial stages. The defect percentage at stage one is 16%. The defect percentage at stage two is 13%. And, the defect percentage at stage three is 10%. Use 3 decimals for probabilities in the following situations. (a)[2] Situation A: the connection logic of the three stages is that a good overall outcome only happens if all three stages individually have good outcomes. Draw the event tree for this situation. Calculate the probability of defective overall outcomes and the probability of good overall outcomes. (b)[2] Situation B: the connection logic of the three stages is that a good overall outcome happens when at least one stage has good outcome. Draw the event tree for this situation. Calculate the probability of defective overall outcomes and the probability of good overall outcomes. (c)[3] Situation C: the connection logic of the three stages is that a good overall outcome happens when at least two stages individually have good outcomes. Draw the event tree for this situation. Calculate the probability of defective overall outcomes and the probability of good overall outcomes. (d)[3] Give some possible real‐life processes for the three situations above.

Answer 1

a) Situation A: the connection logic of the three stages is that a good overall outcome only happens if all three stages individually have good outcomes.

The event tree for this situation:

Probability of good overall outcomes = P(S1)' * P(S2)' * P(S3)'

= 0.84 * 0.87 * 0.90 = 0.67772

Probability of defective overall outcomes = 1 - P(All good)

= 1 - 0.67772 =  0.34228

b) Situation B: the connection logic of the three stages is that a good overall outcome happens when at least one stage has good outcome.

Atleast one stage has good outcome = 1- none has good outcome

The event tree for this situation:

Probability of defective overall outcome= P(S1) * P(S2) * P(S3)

= 0.16 * 0.13 * 0.10 = 0.00208

Probabiliti of good overall outcomes = 1- 0.00208 = 0.99792

c) Situation C: the connection logic of the three stages is that a good overall outcome happens when at least two stages individually have good outcomes.

The event tree for this situation:

Probability of good overall outcomes = P(2 good outcomes) +P(3 good)

= [ P(S1) * P(S2) * P(S3)' +  P(S1) * P(S2)' * P(S3) +P(S1)' * P(S2) +P(S3) ] + P(S1)*P(S2)*P(S3)

= [ 0.84*0.87*0.10 + 0.84*0.13*0.90 + 0.16*0.87*0.90 ] + 0.84*0.87*0.90

= 0.95436

Probability of defective overall outcomes = 1- 0.95436 = 0.04564

d) All three good - A health check up

Atleast one is good - Drug test or security test

Atleast two events needs to be good - No appropriate example can be given. IELTS can be considered. 2 sections performance can affect the overall performance or any examination with a choice.