Download Probability

Probability
Toolbox of Probability Rules
Event
• 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 Joe Pa wins more than 323
games as head coach.
• C = Event that a fraternity is raided next
weekend.
Probability
• Number between 0 and 1, inclusive, that
indicates how likely event is to occur.
• An event with probability of 0 is a null
event, denoted as .
• An event with probability of 1 is a certain
event, denoted as .
• Closer to 1, more likely event is to happen.
• Probability of event A denoted as P(A).
Examples: Null Events
• Man gets pregnant.
• Woman dies of prostate cancer.
Examples: Certain Events
• Sun will set tonight.
• Semester will end.
• Person will die.
3 Ways of Assigning Probabilities
to Events
• Frequentist approach
• Classical approach
• Personal opinion approach
Frequentist Approach
• If an experiment is repeated n times under
essentially identical conditions, and if the
event A occurs m times, then as n grows
large the ratio of m/n approaches a fixed
limit, namely, the probability of A.
Examples: Frequentist Approach
Tosser
#(Tosses)
#(Heads)
P(H)
Buffon
4,040
2,048
0.5069
Pearson
24,000
12,012
0.5005
Kerrich
10,000
5,067
0.5067
3 Ways of Assigning Probabilities
to Events
• Frequentist approach
• Classical approach
• Personal opinion approach
Tool 1
• The complement of an event A, denoted
AC, is the event “not A.”
• 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.
Tool 2
• The intersection of two events A and B,
denoted A  B, is the event “both A and B.”
• 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.
• If two events are independent, then
P(A  B) = P(A)  P(B).
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  B) = 0.36 × 0.45 = 0.16
• 16% of students own bike and have SO.
Tool 3
• The union of two events A and B, denoted
A  B, is the “event A or B, or both.”
• Two events that cannot happen at the same
time are called mutually exclusive.
• If two events are mutually exclusive, then
P(A  B) = P(A) + P(B).
• If two events are not mutually exclusive,
then P(A  B) = P(A) + P(B) - P(A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 B) = P() = 0
• P(A  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
• P(A  B) = 0.37
• So, P(A  B) = 0.75 + 0.38 - 0.37 = 0.76
Tool 4
• The conditional probability of event B
given A has already occurred is denoted
P(B|A).
• P(B|A) = P(A  B)  P(A)
• P(A|B) = P(A  B)  P(B)
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
• Given P(A  B) = 0.17
• P(B|A) = 0.17 ÷ 0.36 = 0.47
• P(A|B) = 0.17 ÷ 0.45 = 0.38
Tool 5
• (Tool 2) Two events are independent if and
only if P(A|B) = P(A) and P(B|A) = P(B).
• (Tool 4) P(A  B) = P(A) × P(B|A)
• (Tool 4) P(A  B) = P(B) × P(A|B)
• (Tool 2) If two events are independent, then
P(A  B) = P(B) × P(A) = P(A) × P(B)