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Chapter 8: Problem 56

The Blue Sky Flight Insurance Company insures passengers against air disasters, charging a prospective passenger \(\$ 20\) for coverage on a single plane ride. In the event of a fatal air disaster, it pays out \(\$ 100,000\) to the named beneficiary. In the event of a nonfatal disaster, it pays out an average of \(\$ 25,000\) for hospital expenses. Given that the probability of a plane's crashing on a single trip is \(.00000087,{ }^{32}\) and that a passenger involved in a plane crash has a \(.9\) chance of being killed, determine the profit (or loss) per passenger that the insurance company expects to make on each trip. HINT [Use a tree to compute the probabilities of the various outcomes.]

Short Answer

Expert verified
The expected profit (or loss) for the Blue Sky Flight Insurance Company per passenger on each trip is approximately \(\$20 - [(\$100,000) \times (87 \times 10^{-9} 脳 0.9)] - [(\$25,000) \times (87 \times 10^{-9} \times 0.1)]\).

Step by step solution

01

Set up the tree diagram

We have three possible outcomes: 1. No disaster/crash 2. Fatal disaster/crash (passenger killed) 3. Nonfatal disaster/crash (passenger survives) Let's denote the following by variables, - P(disaster) 鈥� Probability of a plane crash = .00000087, which is 87 x \(10^{-9}\) - P(fatal) 鈥� Probability of a passenger being killed in a crash = .9 - P(nonfatal) 鈥� Probability of a passenger surviving a crash = .1 Now let's set up the tree: 1. No disaster (probability = 1 - P(disaster)) 2. Disaster 1. Fatal (probability = P(fatal)) 2. Nonfatal (probability = P(nonfatal)) To calculate the probabilities along the branches, we multiply the probabilities of the individual events.
02

Calculate the probabilities of outcomes

Calculate the probabilities of each outcome by following the tree: 1. No disaster: \(1 - 87 \times 10^{-9}\) 2. Disaster 1. Fatal: \(87 \times 10^{-9} \times 0.9\) 2. Nonfatal: \(87 \times 10^{-9} \times 0.1\)
03

Calculate the expected payout

Now we need to find the expected payout for each outcome: 1. No disaster: Payout: $0 2. Disaster 1. Fatal: Payout: \(\$100,000\) 2. Nonfatal: Payout: \(\$25,000\) We can find the expected payout by multiplying the payout by the probability for each outcome and sum them: Expected payout = \(0 + [(\$100,000) \times (87 \times 10^{-9} 脳 0.9)] + [(\$25,000) \times (87 \times 10^{-9} \times 0.1)]\)
04

Calculate the expected profit (or loss) for the insurance company

To calculate the expected profit (or loss) for the insurance company, subtract the cost of insurance coverage from the expected payout: Expected profit (or loss) = expected payout - insurance coverage cost Expected profit (or loss) = \(0 + [(\$100,000) \times (87 \times 10^{-9} 脳 0.9)] + [(\$25,000) \times (87 \times 10^{-9} \times 0.1)]- \$20\) After calculation, we get the expected profit (or loss) for the insurance company.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Tree Diagrams in Probability
Tree diagrams are a helpful tool to visualize all possible outcomes of a probabilistic scenario. In the context of insurance calculations for air travel, tree diagrams make it easy to lay out different potential outcomes for a single flight.

To create a tree diagram, start by listing all initial events鈥攊n our case, whether there's no disaster or a disaster occurs. For each disaster scenario, further branch into more specific outcomes: fatal or nonfatal. Each branch represents a path with a certain probability. Consequently, the complete diagram provides a clear picture of every possible result and their likelihoods.

The usefulness of a tree diagram lies in its simplicity. It helps you calculate the total probability for each outcome by multiplying along the branches. So for our scenario, to find the probability of a fatal disaster, you multiply the probability of a crash by the probability of it being fatal. This organized visualization helps in more complex insurance calculations, making the process less daunting and more intuitive.
Calculating Expected Value
Expected value is a central concept in probability that provides a way to determine the average outcome of a random event over time. It is calculated by multiplying each possible outcome by its probability and then summing all these products.

In our insurance example, the expected payout is calculated by considering all possible financial outcomes from a plane crash or no disaster. There are three main outcomes: no disaster with no payout, a fatal disaster resulting in a large payout, and a non-fatal disaster with a lesser payout.

Expected payout is calculated by:
  • No payout when there is no disaster.
  • The payout amount multiplied by the probability for a fatal crash.
  • The payout amount multiplied by the probability of nonfatal crashes.
By summing these values, the expected value of the payout reflects what the insurance company anticipates paying out over many flights.
Exploring Insurance Calculations
When an insurance company calculates expected profits or losses, they employ probability and expected value to make informed decisions about their policies. Using our air travel insurance exercise, we consider several factors.

First, the company charges a premium for the insurance鈥�$20 in our scenario. Then, based on the probabilities of different outcomes (fatal or nonfatal disaster), they calculate expected payouts. The expected profit or loss is determined by subtracting this payout from the received premium.

Here鈥檚 the formula for expected profit:
  • Subtract the expected payout (calculated using expected value) from the premium collected.
The goal for the insurance company is to ensure this number remains positive, thereby making a profit. If the expected payout is higher than the premium, it may indicate a potential loss. Understanding these calculations helps insurers decide appropriate premium levels to cover risks adequately.

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Most popular questions from this chapter

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