All in One Place

Everything you need for better grades in university, high school and elementary.

Learn with Ease

Made in Canada with help for all provincial curriculums, so you can study in confidence.

Instant and Unlimited Help

Get the best tips, walkthroughs, and practice questions.

0/1
?
Intros
Lessons
  1. Deriving Bayes' rule
0/5
?
Examples
Lessons
  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.
    1. What is the probability that I will get the job?
    2. Given that I got the job what is the probability that she gave me a good 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.
    1. What is the probability that I draw a red marble?
    2. Suppose that I drew a green marble from a bag. What is the probability that I rolled a 6?
  3. False Positives
    A blood test is 95% effective when diagnosing a diseased person. However this blood test also incorrectly diagnoses a healthy person 5% of the time. If 0.1% of the population actually has this disease, then what is the probability that a person has the disease given that they tested positive?

    I like this, gives you hope if you get diagnosed for a serious disease
    Make note of how many healthy people and how many diseased people get diagnosed. Also make a tree diagram illustrating this.
    Topic Notes
    ?
    Recall:
    • Multiplication Rule: P(A  and  B)=P(B)P(AB)P(A \;and\;B)=P(B) \cdot P(A|B)
    • Conditional Probability: P(BA)P(B|A) == P(A  and  B)P(A)\frac{P(A \;and\; B)}{P(A)}
    • Law of Total Probability: P(A)=P(B1)P(AB1)+P(B2)P(AB2)++P(Bn)P(ABn)P(A)=P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots+P(B_n)P(A|B_n)

    Combining all these equations we get Bayes' Rule:
    P(BA)P(B|A) == P(A  and  B)P(A)=P(B)P(AB)P(A)\frac{P(A \;and\; B)}{P(A)}= \frac{P(B) \cdot P(A|B)}{P(A)}
    =P(B)P(AB)P(B1)P(AB1)+P(B2)P(AB2)++P(Bn)P(ABn)=\frac{P(B) \cdot P(A|B)}{P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots+P(B_n)P(A|B_n)}