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An actuary studying the insurance preferences of automobile owners makes the following conclusions: (i) An automobile owner is twice as likely to purchase collision coverage as disability coverage. (ii) The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage. (iii) 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?
in Data Science by Diamond (66,722 points) | 55 views

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It helps if you draw a picture. Create a rectangle labeled S and two circles inside labeled $C$ and $D$. You should have 4 disjoint regions inside the square: $C$ (and) $D$ complement, $C$ (and) $D$, $C$ complement (and) $D$, $C$ complement (and) $D$ complement.

From the picture, you can see that $P(\text{not}  C \text{and}  \text{not} D)$ is equivalent to $P(C U D^{c}$, so it suffices to know $P(C U D)$ and just do 1 minus that. On that note, You know $P[D] = \sqrt{\dfrac{15}{2}}, P[C] = 2P[D]$. Then the formula

$P(C U D) = P(C) + P(D) - P(C \text{and} D)$ allows you to compute $P(C U D)$. Then the answer is just 1 minus this.

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