L2 - Conditional Probability

Question 1

What are Dependent Events?

Answer 1

• the occurrence of one affects the chances of occurrence of the other. If one event has occurred, the probability of the other event changes (it is higher or lower).

Ex: pick a card from a deck of 52:
A = card is a king or an ace; P(A) = 8/52
B = card is royal; P(B) = 12/52

P(A ∩ B) = 4/52

Question 2

What is conditional Probability?

Answer 2

• the probability of
  A given that B has happened:
• written P(A|B)

Question 3

How is Conditional Probability calculated?

Answer 3

P(A|B) = (P(A ∩ B))/(P(B))

OR

P(B|A) = (P(A ∩ B))/(P(A))

Question 4

If events are independent what is the Conditional Probability?

Answer 4

P(A|B) = P(A)

Note the following important important property:

P(A|B) ≠ P(B|A)

Question 5

For Dependent Event what does the Multiplication Rule become?

Answer 5

P(A ∩ B) = P(A|B) x P(B)

P(A ∩ B) = P(B|A) x P(A)

Question 6

For Independent Events what does the Multiplication Rule become?

Answer 6

P(A ∩ B) = P(A) x P(B)
P(A|B) =P(A)
P(B|A) = P(B)

Using these formulas is the quickest way to figure out if two events are independent

Question 7

What is Bayes Theorem?

Answer 7

P(A|B)= (P(A ∩ B))/(P(B)) = (P(B|A) x P(A))/(P(B))

As (P(A ∩ B)) = P(B|A) x P(A)

AND

P(B|A)= (P(A ∩ B))/(P(A)) = (P(A|B) x P(B))/(P(A))

As (P(A ∩ B)) = P(B|A) x P(B)

Question 8

What is Independence or Independent Events?

Answer 8

Independence has to do with the probability of
occurrence, events can occur at the same time. For
independent events:

P(A|B) ≠ 0 and P(A ∩ B) ≠ 0

Question 9

What are Mutually Exclusive Events?

Answer 9

• events cannot occur at the same time

- P(A|B) = 0 and P(A ∩ B) = 0

Question 10

What is Marginal Probabilities?

Answer 10

main events or headings, the sum

P(male) = 0.22 + 0.24 + 0.06 + 0.08 = 0.6 = 60%
P(Economics) = 0.24+0.26 = 0.5 = 50%. 

• This is the Sum or the various Joint Probabilities e.g. Sum of Male doing Economics plus Male doing Finance + Male doing Banking

Question 11

What are Join Probability?

Answer 11

Joint probability: the cells

P(male ∩ Econ) = 360/1500 = 0.24

Question 12

How can P(A|B) be written as the sum of of joint Probabilities?

Answer 12

P(A|B)= (P(A ∩ B))/(P(B)) = (P(B|A) x P(A))/(P(B))

= (P(B|A) x P(A))/|((P(B|A) x P(A)) + (P(B|notA ) x P(notA)))