Transcript Probability - Penn State Statistics Department

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
AC, is “the event that A does not happen.”
• P(AC) = 1 - P(A)
Example: Tool 1
•
•
•
•
•
Suppose 1% of population is alcoholic.
Let A = event person is alcoholic.
Then AC = event 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.
• 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 = 0.16
• 16% 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) = 0.32 + 0.38 = 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) = 0.75 + 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 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?