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?