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 student has four exams in one day. B = Event PSU football team wins national championship. 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 A C, is “the event that A does not happen.” P(A C ) = 1 - P(A)

Example: Tool 1 Suppose 1% of population is alcoholic. Let A = event person is alcoholic. Then A C = event person is not alcoholic. P(A C ) = = 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. Is it 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. Is it 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 student has significant other. P(B) = 0.45 Assuming bike ownership is independent of having SO: P(A and B) = 0.36 × 0.45 = % of students own bike and have SO.

Example: Tool 2 Let A = event student is male. P(A) = 0.50 Let B = event student is sleep deprived. P(B) = 0.60 A and B = 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 student is drunk. Let B = event student is sober. A or B = event student is either drunk or sober. Are A and B mutually exclusive?

Example to Prelude to Tool 3 Let A = event student is drunk Let B = event student is in love A or B = event 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) = = 0.70

Example: Tool 3 Let A = event randomly selected student drinks alcohol. P(A) = 0.75 Let B = event student ever tried marijuana. P(B) = 0.38 A and B = event student drinks alcohol and has tried marijuana. P(A and B) = 0.37 P(A or B) = = 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 student owns bike, and B = event student has a significant other. P(B|A) is the probability that a student has a significant other “given” (or “if”) he/she owns a bike. P(A|B) is the probability that a 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?