More on Probability Rules (Exam 2)

Question 1

disjoint events

Answer 1

• 2 events that are mutually exclusive, they share no outcomes in common
• Ex: I flip a coin 3 times
  A = I toss no heads {0}
  B = I toss 2+ heads = {2, 3}
• 2 separate circles

Question 2

NOT disjoint

Answer 2

• 2 events that share outcomes in common
• ex: flip a coin 3 times
  A = I toss all or no heads {0, 3}
  B = I toss 2 or fewer heads = {0, 1, 2}
• 2 circles overlap

Question 3

intersection of events

Answer 3

• when there is overlap between events
• not disjoint
• 2 circles overlapping
• the event that occurs when BOTH A & B occur (the gray area where circles overlap in the middle)
• (A & B) = (A ∩ B), “A intersect B” OR “A and B”

Question 4

law of total probability

Answer 4

• P(A) = P(A ∩ B) + P(A ∩ Bc)
• common application of intersections
• If I want to find the chance of event A occurring, I can use a secondary event B, to partition (break down) that first event into components
  ex: P(cat) = P(cat ∩ dog) + P(cat ∩ no dog)

Question 5

multiplication rule for independent events

Answer 5

• when 2 events are independent, the probability of both of them happening is P(A ∩ B) = P(A) x P(B)
• you MUST be certain that they are both independent
• ex: if I flip 2 coins, what is the probability that both coins land on heads? P(both heads) = P(1st heads ∩ 2nd heads) = P(1st heads) x P(2nd heads) = ½ x ½ = ¼

Question 6

independent events

Answer 6

• one event has no effect on the other
• ex: result of 2 coin flips

Question 7

union

Answer 7

• the event that occurs when EITHER A or B occurs
• (A or B) = (A ∩ B)
• when 2 events are being double counted but shouldn’t be

Question 8

union

Answer 8

• the event that occurs when EITHER A OR B occurs
• (A or B) = (A ∩ B)
• when 2 events are being double counted but shouldn’t be

Question 9

general addition rule

Answer 9

• for disjoint events, P(A ⋃ B) = P(A) + P(B) – P(A ∩ B)
• If A & B are disjoint, P(A ∩ B) = 0

Question 10

conditions

Answer 10

additional info

Question 11

conditional

Answer 11

• there are times when we want to know the probability of an event, CONDITIONAL on some other event being occurred
• ex: Probability that my stock’s value increases tomorrow VS. probability that my stock’s value increases tomorrow given that it decreased today
• P(A|B)… evaluate the proportion of times event A occurs only in repetitions of the random phenomenon where event B has occurred

Question 12

Conditioning on an event having occurred typically reduces the size of the sample space/population we’re interested in

Answer 12

ex: Probability that my stock’s value increases tomorrow (all days) VS. Probability that my stock’s value increases tomorrow given that it decreased today (only days where stock value decreased the day before)