STAT1008 Lecture Notes - Lecture 11: Conditional Probability, Random Variable, Dependent And Independent Variables
PROBABILITY RULES 11.1
Outline
- Probabilities for equally likely outcomes
- Compement rule: P (not A)
- Additive rule: P (A or B)
- Conditional probability: P (B if A)
o Multiplicative rule: p(A and B)
o Special cases:
▪ Independent
▪ Mutually exclusive
Event
- An event is something that either happens or does’t happe, o soethig that eithe is tue or is not
true
- Examples:
o A randomly selected card is a Heart
o The response variable Y > 90
o A randomly selected person is male
o It rains today
Probability
- The probability of event A, P(A), is the long run frequency or proportion of times the event occurs.
o Probability is always between 0 and 1
o Probability always refers to an event
o P(A) = 1 means A will definitely happen
o P(A) = 0 means A will definitely not happen
Probability Examples
- Y = number of siblings
o P(Y = 1) = 0.419 (based on survey data)
- For U.S, college students in 2010
o P(Gender = female) = 0.585 (World DataBank)
- P(it rains today) = 0.3)
Equally Likely Outcomes
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Combining Events
- P(A and B) is the probability that both events A and B will
happen
- P(A or B) is the probability that either event A or event B will
happen (or both)
Additive Rule: P(A or B)
Complement rule: P (not A)
Conditional Probability
- P(A if B) is the probability of A, if we know B has
happened
- This is read in multiple ways:
o poaility of A if B
o poaility of A gie B
o poaility of A oditioal o B
- You may also see this written as P(A | B)
- Note: PA if B ≠ PB if A
Multiplicative Rule: P (A and B)
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Summary: Probability Rules
Disjoint Events
- Events A and B are disjoint or mutually exclusive if only one of the two events can happen
- Think of two events that are disjoint, and two events that are not disjoint.
- If A and B are disjoint, then P(A or B) = P(A) + P(B)
o P(A or B) = P(A) + P(B) – P(A and B)
o If A and B are disjoint, then both cannot happen, so P(A and B) = 0.
P(A or B)
Independence
- Events A and B are independent if P(A if B) = P(A).
- Intuitively, knowing that event B happened does not change the probability that event A happened.
- Think of two events that are independent, and two events that are not independent.
- Pick the best choice: If A and B are independent, then
o P(A and B) = P(A)P(B)
o P( A and B) = P(A if B)P(B)
o If A and B are independent, then P(A if B) = P(A), so P(A and B) = P(A)P(B)
P(A and B)
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Document Summary
Conditional probability: p (b if a: multiplicative rule: p(a and b, special cases, mutually exclusive. An event is something that either happens or does(cid:374)"t happe(cid:374), o(cid:396) so(cid:373)ethi(cid:374)g that eithe(cid:396) is t(cid:396)ue or is not true. Examples: a randomly selected card is a heart, the response variable y > 90. The probability of event a, p(a), is the long run frequency or proportion of times the event occurs: probability is always between 0 and 1. Probability always refers to an event: p(a) = 1 means a will definitely happen, p(a) = 0 means a will definitely not happen. Y = number of siblings: p(y = 1) = 0. 419 (based on survey data) For u. s, college students in 2010: p(gender = female) = 0. 585 (world databank) P(a and b) is the probability that both events a and b will happen. P(a or b) is the probability that either event a or event b will happen (or both)
