Multiplication Rule for Independent Events

Independence

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Multiplication

$A, B$ are independent iff:

P (A \cap B) = P (A) \cdot P (B)

Proof
Using the Multiplication Rule for Conjunct, then $P (A \cap B) = P (A | B) \cdot P (B)$ which becomes $P (A) \cdot P (B)$ when $P (A | B) = P (A)$ during independence.
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Washing Machines

It is know $.3$ of a company's washing machines break down in the first 10 years of use, while only $.1$ of its dryers do. If someone in NYC purchases a washer by this company, and his brother in SFO buys a dryer made by this company, what is the probability that both machines break down within 10 years?

Proof
Let $A$ denote the event the washer breaks down in the first 10 years, and $B$ for the for dryer. Then $P (A) = .3$ and $P (B) = .1$ . Assuming independent functionality of the machines, then:

P (A \cap B) = .3 \cdot .1 = .03

The probability that neither machine breaks down is:

P (A^{'} \cap B^{'}) = (1 - .3) (1 - .1) = .63

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