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The scheduling officer for a local police department is trying to schedule additional patrol units in each of two neighbourhoods - southern and northern. She knows that on any given day, the probabilities of major crimes and minor crimes being committed in the northern neighbourhood were 0.418 and 0.612, respectively, and that the corresponding probabilities in the southern neighbourhood were 0.355 and 0.520. Assuming that all crime occur independent of each other and likewise that crime in the two neighbourhoods are independent of each other, what is the probability that no crime of either type is committed in either neighbourhood on any given day?
For northern neighbourhood,
Probability that there is no major crime = $$(1 - 0.418) = 0.582$$
Probability that there is no minor crime = $$(1 - 0.612) = 0.388$$
For southern neighbourhood,
Probability that there is no major crime = $$(1 - 0.355) = 0.645$$
Probability that there is no minor crime = $$(1 - 0.520) = 0.480$$
$$\therefore$$ Probability that no crime of either type is committed in either neighbourhood on any given day
= $$0.582 \times 0.388 \times 0.645 \times 0.480$$
= $$0.069$$
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