A car insurance company insures four types of drivers:

• Good drivers who have a 3% chance of getting in an accident each year
• Mediocre drivers have a 7% chance of getting in an accident in each year
• Atrocious drivers who have a 20% chance of getting in an accident in each year
• Ludicrously bad drivers who have a 75% chance of getting in an accident each year 1

70% of drivers are good, 20% are mediocre, 7% are atrocious, and 3% are ludicrously bad. A driver they insure gets in an accident, what is the likelihood he is a good, mediocre, atrocious, or ludicrously bad driver?

A driver they insure gets in an accident, the likelihood he is

A good driver, P(G|GA) = 0.294

A mediocre driver, P(M|GA) = 0.196

A atrocious driver, P(A|GA) = 0.196

A ludicrously bad driver, P(L|GA) = 0.315

Step-by-step explanation:

Let G represent good drivers, P(G) = 0.70

M represent mediocre drivers, P(M) = 0.20

A represent Atrocious drivers, P(A) = 0.07

L represent Ludicrously bad drivers, P(L) = 0.03

If we represent the probability of getting involved in an accident as P(GA),

We're given in the question that the probability of getting in an accident, given one is a good driver, P(GA|G) = 0.03

P(GA|M) = 0.07

P(GA|A) = 0.20

P(GA|L) = 0.75

And we're told to find the probability that given an accident has occurred, the driver is a good driver, P(G|GA), moderate driver, P(M|GA), atrocious driver, P(A|GA) and ludicrously bad driver, P(L|GA).

To do this, we will require the total probability of getting in an accident P(GA) which is the total of probability of each type of driver getting in an accident.

P(GA) = P(G n GA) + P(M n GA) + P(A n GA) + P(L n GA)

To obtain P(G n GA)

Conditional probability P(GA|G) given in the question is represented mathematically as P(GA n G)/P(G).

So, P(GA n G) = P(G n GA) = P(GA|G) × P(G) = 0.7 × 0.03 = 0.021

P(GA n M) = P(M n GA) = P(GA|M) × P(M) = 0.07 × 0.2 = 0.014

P(GA n A) = P(A n GA) = P(GA|A) × P(M) = 0.20 × 0.07 = 0.014

P(GA n L) = P(L n GA) = P(GA|L) × P(L) = 0.75 × 0.03 = 0.0225

P(GA) = 0.021 + 0.014 + 0.014 + 0.0225 = 0.0715

So, our required probability now,

P(G|GA) = P(G n GA)/P(GA) = 0.021/0.0715 = 0.294

P(M|GA) = P(M n GA)/P(GA) = 0.014/0.0715 = 0.196

P(A|GA) = P(A n GA)/P(GA) = 0.014/0.0715 = 0.196

P(L|GA) = P(L n GA)/P(GA) = 0.0225/0.0715 = 0.315

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step-by-step explanation:

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