Driver Devon Cooper has a poor driving record and pays...

Driver Devon Cooper has a poor driving record and pays double the usual premium as a high-risk driver. The regular premium would be $$\$ 490$$ for a year. If Devon must pay the high-risk premium every year for 5 years, how much more will he pay for insurance premiums than a low-risk driver receiving a $10 \%$ discount over the same 5-year period?.________

Key Concepts

Insurance Premium Calculation

This concept involves determining the cost of an insurance policy based on base rates, which can be adjusted according to various factors. It is crucial for understanding how premiums are set and how changes, such as doubling for high-risk drivers, influence overall costs.

Risk-Based Pricing

Risk-based pricing adjusts insurance premiums based on the policyholder’s perceived risk level. Higher risk profiles lead to higher premiums, reflecting the increased likelihood and potential cost of claims for that driver.

Percentage Discounts

Percentage discounts are reductions applied to a base amount, calculated by reducing the base by a given percentage. In insurance, low-risk drivers may receive such discounts, resulting in a lower premium compared to the standard rate.

Multiplicative Cost Calculations Over Time

This concept involves calculating total costs over a given time period by multiplying the periodic cost (such as a yearly premium) by the number of periods. It is essential for comparing costs over multiple years, enabling individuals to assess long-term financial impacts.

Transcript

00:01 So in this question, we have s is the event that you're a safe driver.

00:09 So sc is going to be the event that you're an unsafe driver.

00:16 And we have the probability of having an accident.

00:19 So let's say that a is the event that you have an accident, and ac is the event that you have no accident.

00:29 We're told that 70 % of the population are safe.

00:32 So ps is 0 .7.

00:34 And if you're safe, you have a probability of 0 .1 of causing an accident.

00:44 But if you're not safe, then the probability of having an accident is 0 .5.

00:54 Now the insurance premium is $400 times the probability of having an accident in the following year for a particular person.

01:14 Now a new subscriber has an accident during the first year.

01:19 So what should their insurance premium be for the next year? so what's the probability of having an accident in the second year, given that you have an accident in the first year? well, that's going to be the probability of having an accident in the second year, given that you're safe, times the probability that you're safe, given that you had an accident in the first year, plus the probability of having an accident in the second year, given that you're not safe, times the probability that you're not safe, given that you had an accident in the first year.

01:58 So we need to work out, what are the probabilities of being safe or unsafe, given that you had an accident? well, the probability of being safe, given that you have an accident, is the probability of having an accident, given that you're safe, the probability of being safe, over the probability of having an accident, which we can decompose in this way...

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