Respuesta :
Answer:
a) 0.035 = 3.5% probability that a customer is a good risk and has filed a claim.
b) 0.0395 = 3.95% probability that the customer has filed a claim.
c) 0.8861 = 88.61% probability that the customer is a good risk
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
a) What is the probability that the customer is a good risk and has filed a claim?
70% are good risks.
Of those, 0.5% file a claim. So
[tex]0.7*0.05 = 0.035[/tex]
0.035 = 3.5% probability that a customer is a good risk and has filed a claim.
b) What is the probability that the customer has filed a claim?
0.5% of 70%(good risks)
1% of 20%(medium risks)
2.5% of 10%(poor risks). So
[tex]0.05*0.7 + 0.01*0.2 + 0.025*0.1 = 0.0395[/tex]
0.0395 = 3.95% probability that the customer has filed a claim.
c) Given that the customer has filed a claim, what is the probability that the customer is a good risk?
0.0395 = 3.95% probability that the customer has filed a claim means that [tex]P(A) = 0.0395[/tex]
0.035 = 3.5% probability that a customer is a good risk and has filed a claim means that [tex]P(A \cap B) = 0.035[/tex]
Thus
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.035}{0.0395} = 0.8861[/tex]
0.8861 = 88.61% probability that the customer is a good risk
