Question

The probability that a homeowners policyholder reports a property claim in a year increases by 25% per year. Conversely, the probability that a homeowners policyholder reports a liability claim in a year decreases by 25% per year.

The probability that a homeowners policyholder reports both a property claim and a liability claim in Year 1 is 0.01. The event that a homeowners policyholder reports a property claim is independent of the event that the policyholder reports a liability claim.

Calculate the probability that a homeowners policyholder reports both a property claim and a liability claim in Year 9.

Answer 1

To calculate the combined probability of both property and liability claims in Year 9, determine the separate probabilities at that time and multiply them, since the events are independent. The initial probability of 0.01 is adjusted by the respective annual increases and decreases over 8 years to find the probabilities in Year 9.

Step-by-step explanation:

The question asks to calculate the probability that a homeowners policyholder reports both a property claim and a liability claim in Year 9, given that the probability of a property claim increases by 25% each year and the probability of a liability claim decreases by 25% each year. Additionally, the events of reporting property and liability claims are independent and the initial probability in Year 1 is 0.01.

To find the probability of both events occurring in Year 9, we need to determine the probabilities of each event occurring separately at that time, and then multiply them, since the events are independent. If we denote the initial probabilities of the property and liability claims as P and L respectively, and the probabilities in Year 9 as P9 and L9, we can use the formula for an increasing probability P9 = P × (1+0.25)^8 and a decreasing probability L9 = L × (1-0.25)^8. After finding P9 and L9, the combined probability in Year 9 will be P9 × L9.