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# Conditional Probability

Probability and Statistics
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Probability Theory
schedule Mar 9, 2022
Last updated
local_offer
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Let $A$ and $B$ be any two mutually exclusive events defined over $S$. Then the following is true:

$$\mathbb{P}(B|A)=\frac{\mathbb{P}(B\cap{A})}{\mathbb{P}(A)}$$

$\mathbb{P}(B|A)$ is read as the "probability of $B$ given that $A$ occurred".

# Example

Question. The probability that a product is defective is 0.5. The probability that a product is defective and a customer complains is 0.2. What is the probability that a customer complains given that the product is defective?

Solution. Let A be the event that the product is defective. Let B denote the event that a customer complains. Then probability that a customer complains given that the product is defective is:

$$\mathbb{P}(B|A)=\frac{\mathbb{P}(B\cap{A})}{\mathbb{P}(A)} =\frac{0.2}{0.5} =0.4$$
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