The probability that a Kaduna businessman goes to Lagos by car is 0.6, and by air is 0.4. If he goes by car, the probability that he will be on time for his appointment is 0.3 and if he goes by air it is 0.65
i)Find the probability that he arrives in Lagos early for an appointment
ii)One Monday, he arrived late for his appointment. Find the probability that he went by car?
Make yourself a tree-diagram with two main branches C --- by car and A --- by airplane
Each of those will have 2 branches of O -- on time and L -- for late
Enter the probability along each of the branches, so
CO ---> (.6)(.3) = .18
CL ---> (.6)(.7) = .42
AO ---> (.4)(.65) = .26
AL ---> (.4)(.35) = .14
Note they add up to 1 , as expected
i) add CO + A)
ii) that would be AL
Yes
To find the probability that the businessman arrives in Lagos early for an appointment, we need to consider two cases: if he goes by car and if he goes by air.
i) Probability that he arrives early if he goes by car:
Given: P(Goes by car) = 0.6 and P(On time | Goes by car) = 0.3
Therefore, P(Arrives early | Goes by car) = 1 - P(On time | Goes by car) = 1 - 0.3 = 0.7
ii) Probability that he arrives early if he goes by air:
Given: P(Goes by air) = 0.4 and P(On time | Goes by air) = 0.65
Note that the problem does not provide the probability of arriving early if he goes by air. In this case, we cannot calculate the exact probability without additional information.
For the second question, we are given that he arrived late for his appointment on a Monday.
Let's assume that there are only two options: going by car (C) or going by air (A).
To find the probability that he went by car given that he arrived late (P(C | Late)), we need to use Bayes' theorem:
P(C | Late) = (P(Late | C) * P(C)) / P(Late)
Given:
P(Late | C) = 1 - P(On time | C) = 1 - 0.3 = 0.7
P(C) = 0.6
P(Late) = (P(Late | C) * P(C)) + (P(Late | A) * P(A))
= (0.7 * 0.6) + ? *(Unknown probability when he goes by air)
Since we don't know the probability of arriving late if he goes by air, we cannot calculate P(Late) or P(C | Late) without this information.