probability toolbox of probability rules. event an event is the result of an observation or...
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Probability
Toolbox of Probability Rules
Event
• An event is the result of an observation or experiment, or the description of some potential outcome.
• Denoted by uppercase letters: A, B, C, …
Examples: Events
• A = Event President Clinton is impeached from office.
• B = Event PSU men’s basketball team gets lucky and wins their next game.
• C = Event that a fraternity is raided next weekend.
Notation: The probability that an event A will occur is denoted as P(A).
Tool 1
• The complement of an event A, denoted AC, is “the event that A does not happen.”
• P(AC) = 1 - P(A)
Example: Tool 1
• Assume 1% of population is alcoholic.
• Let A = event randomly selected person is alcoholic.
• Then AC = event randomly selected person is not alcoholic.
• P(AC) = 1 - 0.01 = 0.99
• That is, 99% of population is not alcoholic.
Prelude to Tool 2
• The intersection of two events A and B, denoted “A and B”, is “the event that both A and B happen.”
• Two events are independent if the events do not influence each other. That is, if event A occurs, it does not affect chances of B occurring, and vice versa.
Example for Prelude to Tool 2
• Let A = event student passes this course
• Let B = event student gives blood today.
• The intersection of the events, “A and B”, is the event that the student passes this course and the student gives blood today.
• Do you think it is OK to assume that A and B are independent?
Example for Prelude to Tool 2
• Let A = event student passes this course
• Let B = event student tries to pass this course
• The intersection of the events, “A and B”, is the event that the student passes this course and the student tries to pass this course.
• Do you think it is OK to assume that A and B are not independent, that is “dependent”?
Tool 2
• If two events are independent, then P(A and B) = P(A) P(B).
• If P(A and B) = P(A) P(B), then the two events A and B are independent.
Example: Tool 2
• Let A = event randomly selected student owns bike. P(A) = 0.36
• Let B = event randomly selected student has significant other. P(B) = 0.45
• Assuming bike ownership is independent of having SO: P(A and B) = 0.36 × 0.45 = 0.16
• 16% of students own bike and have SO.
Example: Tool 2
• Let A = event randomly selected student is male. P(A) = 0.50
• Let B = event randomly selected student is sleep deprived. P(B) = 0.60
• A and B = randomly selected student is sleep deprived and male. P(A and B) = 0.30
• P(A) × P(B) = 0.50 × 0.60 = 0.30
• P(A and B) = P(A) × P(B). So, being male and being sleep-deprived are independent.
Prelude to Tool 3
• The union of two events A and B, denoted A or B, is “the event that either A happens or B happens, or both A and B happen.”
• Two events that cannot happen at the same time are called mutually exclusive events.
Example to Prelude to Tool 3
• Let A = event randomly selected student is drunk.
• Let B = event randomly selected student is sober.
• A or B = event randomly selected student is either drunk or sober.
• Are A and B mutually exclusive?
Example to Prelude to Tool 3
• Let A = event randomly selected student is drunk
• Let B = event randomly selected student is in love
• A or B = event randomly selected student is either drunk or in love
• Are A and B mutually exclusive?
Tool 3
• If two events are mutually exclusive, then P(A or B) = P(A) + P(B).
• If two events are not mutually exclusive, then P(A or B) = P(A)+P(B)-P(A and B).
Example: Tool 3
• Let A = randomly selected student has two blue eyes. P(A) = 0.32
• Let B = randomly selected student has two brown eyes. P(B) = 0.38
• P(A or B) = 0.32 + 0.38 = 0.70
Example: Tool 3
• Let A = event randomly selected student does not abstain from alcohol. P(A) = 0.75
• Let B = event randomly selected student ever tried marijuana. P(B) = 0.38
• A and B = event randomly selected student drinks alcohol and has tried marijuana.
• P(A and B) = 0.37• P(A or B) = 0.75 + 0.38 - 0.37 = 0.76
Tool 4
• The conditional probability of event B given A has already occurred, denoted P(B|A), is the probability that B will occur given that A has already occurred.
• P(B|A) = P(A and B) P(A)
• P(A|B) = P(A and B) P(B)
Example: Tool 4
• Let A = event randomly selected student owns bike, and B = event randomly selected student has significant other.
• P(B|A) is the probability that a randomly selected student has a significant other “given” (or “if”) he/she owns a bike.
• P(A|B) is the probability that a randomly selected student owns a bike “given” he/she has a significant other.
Example: Tool 4
• Let A = event randomly selected student owns bike. P(A) = 0.36
• Let B = event randomly selected student has significant other. P(B) = 0.45
• P(A and B) = 0.17
• P(B|A) = 0.17 ÷ 0.36 = 0.47
• P(A|B) = 0.17 ÷ 0.45 = 0.38
Tool 5
• Alternative definition of independence: – two events are independent if and only if P(A|
B) = P(A) and P(B|A) = P(B).
• That is, if two events are independent, then P(A|B) = P(A) and P(B|A) = P(B).
• And, if P(A|B) = P(A) and P(B|A) = P(B), then A and B are independent.
Example: Tool 5
• Let A = event student is female
• Let B = event student abstains from alcohol
• P(A) = 0.50 and P(B) = 0.12
• P(A|B) = 0.50 and P(B|A) = 0.12
• Are events A and B independent?
Example: Tool 5
• Let A = event student is female
• Let B = event student dyed hair
• P(A) = 0.50 and P(B) = 0.40
• P(A|B) = 0.65 and P(B|A) = 0.52
• Are events A and B independent?
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