Bayes' rule  Probability
Bayes' rule
Lessons
Notes:
Recall:
• Multiplication Rule: $P(A \;and\;B)=P(B) \cdot P(AB)$
• Conditional Probability: $P(BA)$ $=$ $\frac{P(A \;and\; B)}{P(A)}$
• Law of Total Probability: $P(A)=P(B_1)P(AB_1)+P(B_2)P(AB_2)+ \cdots+P(B_n)P(AB_n)$
Combining all these equations we get Bayes’ Rule:
$P(BA)$ $=$ $\frac{P(A \;and\; B)}{P(A)}= \frac{P(B) \cdot P(AB)}{P(A)}$
$=\frac{P(B) \cdot P(AB)}{P(B_1)P(AB_1)+P(B_2)P(AB_2)+ \cdots+P(B_n)P(AB_n)}$

1.
Bayes' Rule
I am going to ask my boss to be my reference after applying to another job. If she gives me a good recommendation there is a 0.75 probability that I will get the job. While if she gives me a bad recommendation there is only a 0.25 probability that I will get the job. There is a 60% chance she will give me a good reference and a 40% chance she will give me a bad reference. 
2.
I have 3 bags that each contains 5 marbles.
Bag A:
Bag B:
Bag C:
2 Green
4 Green
5 Green
3 Red
1 Red
0 Red
I roll a fair die to decide which bag I will draw from. If I roll a 1,2,3 I will draw a marble from Bag A. If I roll a 4,5 I will draw from Bag B. And if I roll a 6, then I will draw a marble from bag C.