# Class Conditional Probability Function

## References

### 2011

• (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Conditional_probability#Example
• Consider the rolling of two fair six-sided dice.
• Let $\displaystyle{ A }$ be the value rolled on die 1
• Let $\displaystyle{ B }$ be the value rolled on die 2
• Let $\displaystyle{ A_n }$ be the event that $\displaystyle{ A=n }$
• Let $\displaystyle{ \Sigma_m }$ be the event that $\displaystyle{ A+B \leq m }$
• … Suppose however we roll the dice many times, but ignore cases in which $\displaystyle{ A+B\gt 5 }$. In what proportion of the remaining rolls would $\displaystyle{ A=2 }$? … $\displaystyle{ A=2 }$ in 3 of these. The answer is therefore $\displaystyle{ \textstyle \frac{3}{10} = 0.3 }$. We say, the probability that $\displaystyle{ A=2 }$ given that $\displaystyle{ A+B \leq 5 }$, is 0.3. This is a conditional probability, because it has a condition that limits the sample space. In more compact notation, $\displaystyle{ P(A_2 | \Sigma_5) = 0.3 }$.