An insurance company believes that 30% of drivers are careless, and that the probability of a driver having an accident in any one year is 0.4 for a careless driver and 0.2 for a careful driver. Find the probability that a driver will have an accident next year given that she has had an accident this year.
my lecturer used
P(C/A) * P(A|C) + P(C'/A) * P(A|C')
i dont understand why, please help me
30% drivers are careless , P(C) = 0.3
P(A|C) = probability that a person has accident given that they are careless = 0.4
Similarly P(A|C') = 0.2 [Where C' is non careless drivers]
Now she could be a careless OR careful driver who had accident last year.
Probability that she is careless given that she had and accident= P(C|A)
Probability that she will meet accident next year if she is careless = P(A|C)
Therefore, Probability that she will meet accident next year if she is careless AND had an accident last year = P(A|C)P(C|A)
If you understood this part then you will be able to solve the case where she is NOT careless
Remember we used OR in the beginning so you should add those cases to get,
P = P(C|A)P(A|C) + P(C'|A)P(A|C')
Hope this helps.