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Do The Math

An actuary studying the insurance preferences of automobile owners makes the following conclusions:

  1. An automobile owner is twice as likely to purchase collision coverage as disability coverage.
  2. The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage.
  3. The probability that an automobile owner purchases both collision and disability coverages is 0.15.

What is the probability that an automobile owner purchases neither collision nor disability coverage?

  1. 0.18
  2. 0.33
  3. 0.48
  4. 0.67
  5. 0.82

Solution: B

Let

    C = Event that a policyholder buys collision coverage
    D = Event that a policyholder buys disability coverage

Then we are given that

By the independence of C and D, it therefore follows that

Now the independence of C and D also implies the independence of CC and DC. As a result, we see that