Download Probability - NCSU Statistics

Probability
•Formal study of uncertainty
•The engine that drives statistics
• Primary objective of lecture unit 4: use
the rules of probability to calculate
appropriate measures of uncertainty.
Introduction
• Nothing in life is certain
• We gauge the chances of successful
outcomes in business, medicine, weather,
and other everyday situations such as the
lottery (recall the birthday problem)
History
• For most of human history, probability, the
formal study of the laws of chance, has
been used for only one thing: gambling
History (cont.)
• Nobody knows exactly when gambling
began; goes back at least as far as ancient
Egypt where 4-sided “astragali” (made from
animal heelbones) were used
History (cont.)
• The Roman emperor Claudius (10BC-54AD)
wrote the first known treatise on gambling.
• The book “How to Win at Gambling” was
lost.
Rule 1: Let Caesar win IV
out of V times
Approaches to Probability
• Relative frequency
event probability = x/n,
where x=# of occurrences of event of
interest, n=total # of observations
• Coin, die tossing; nuclear power plants?
• Limitations
repeated observations not practical
Approaches to Probability (cont.)
• Subjective probability
individual assigns prob. based on personal
experience, anecdotal evidence, etc.
• Classical approach
every possible outcome has equal
probability (more later)
Basic Definitions
• Experiment: act or process that leads to a
single outcome that cannot be predicted
with certainty
• Examples:
1. Toss a coin
2. Draw 1 card from a standard deck of cards
3. Arrival time of flight from Atlanta to RDU
Basic Definitions (cont.)
• Sample space: all possible outcomes of an
experiment. Denoted by S
• Event: any subset of the sample space S;
typically denoted A, B, C, etc.
Simple event: event with only 1 outcome
Null event: the empty set F
Certain event: S
Examples
1. Toss a coin once
S = {H, T}; A = {H}, B = {T} simple events
2. Toss a die once; count dots on upper face
S = {1, 2, 3, 4, 5, 6}
A=even # of dots on upper face={2, 4, 6}
B=3 or fewer dots on upper face={1, 2, 3}
Laws of Probability
1. 0  P ( A)  1, for any event A
2. P (F )  0, P ( S )  1
Laws of Probability (cont.)
3.
P(A’ ) = 1 - P(A)
For an event A, A’ is the complement of A; A’
is everything in S that is not in A.
S
A'
A
Birthday Problem
• What is the smallest number of people you
need in a group so that the probability of 2
or more people having the same birthday is
greater than 1/2?
• Answer: 23
No. of people
23 30 40 60
Probability
.507 .706 .891 .994
Example: Birthday Problem
• A={at least 2 people in the group have a
common birthday}
• A’ = {no one has common birthday}
3 people
23 people
:P ( A') 
364
363

365
365
:
364
363
343
P ( A') 


 . 498
365
365
365
so P ( A )  1  P ( A ' )  1  . 498  . 502
Unions and Intersections
S
A
B
A
A
Mutually Exclusive (Disjoint) Events
• Mutually exclusive or disjoint events-no
outcomes from S in common
A = 
S
A
B
Laws of Probability (cont.)
Addition Rule for Disjoint Events:
4. If A and B are disjoint events, then
P(A  B) = P(A) + P(B)
Laws of Probability (cont.)
General Addition Rule
5. For any two events A and B
P(A  B) = P(A) + P(B) – P(A  B)
P(AB)=P(A) + P(B) - P(A B)
S
A
A
B
Example: toss a fair die once
•
•
•
•
S = {1, 2, 3, 4, 5, 6}
A = even # appears = {2, 4, 6}
B = 3 or fewer = {1, 2, 3}
P(A B) = P(A) + P(B) - P(A B)
=P({2, 4, 6}) + P({1, 2, 3}) - P({2})
= 3/6 + 3/6 - 1/6 = 5/6
Laws of Probability: Summary
•
•
•
•
1. 0  P(A)  1 for any event A
2. P() = 0, P(S) = 1
3. P(A’) = 1 – P(A)
4. If A and B are disjoint events, then
P(A  B) = P(A) + P(B)
• 5. For any two events A and B,
P(A  B) = P(A) + P(B) – P(A  B)
End of First Part of Section 4.1