Download Chapter 6. Rules of Probability Events and the Venn Diagram The

Chapter 6. Rules of Probability
Events and the Venn Diagram
„ Probabilities can be added up, subtracted,
multiplied and divided – in a somewhat
different sense than what we know.
„ Some basic terminologies:
„ Event: is a subset of a sample space.
„ Experiment: a process of observation or
measurement;
„ Outcome of an experiment;
„ Sample space: the set of all possible
outcomes; it is usually denoted by S;
„ Examples: flip a coin once: S = {H,T}; flip a
coin twice: S = {HH,HT,TH,TT}; …
„ Examples: flip a coin twice, all heads up; throw
a die once, the number is odd, …
„ Events are denoted by capital letters: E, A,…
„ The Venn diagram
S
E
or
E
The Operations of Events
The Operations of Events
„ For two events A and B from a sample space
E, we can consider
„ Mutually exclusive events: if A∩B = Ø, i.e., no
„ Union: AUB which consists of all elements
contained in A, in B or in both;
„ Intersection: A∩B which consists of all the
elements contained in both A and B;
„ Difference: A\B which consists of all the
elements contained in A but not in B;
S
A
B
element is in common to A and B;
„ Collectively exhaustive events: if AUB = S,
i.e., A and B constitute the entire sample
space.
„ Complement events: if both A∩B = Ø and
AUB = S;
A = B’ and B = A’
A
A
B
B
A
B
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De Morgan’s Rule
Operations of Probability
„ (AUB)′ = A′∩B′ and (A ∩ B)′ = A′ UB′;
„ In Venn diagram,
=
AUB
A
A′
„ The classical definition of probability: for any
Ex: A = I go;
B = you go
∩
B′
B
event E in the sample space S, the probability
for E to occur is
The number of outcomes in E
P (E ) =
The number of outcomes in S
The area of E
=
The area of S
„ For any E in S, 0 ≤ P(E) ≤ 1;
„ P(Ø) = 0 and P(S) = 1;
The Frequency Interpretation of Probability
and the Law of Large Numbers
A Special Addition Rule
„ The probability of an event is the proportion
„ If A and B are mutually exclusive, then
of the time that events of the same kind will
occur in the long run.
„ The law of large numbers: If a trial is
repeated a large number of times, the
proportion of successes will tend to approach
the probability that any one will be a success.
Example: the chance of wining a lottery.
„ It is necessary to count the number of all
possible outcomes as well as the number of
“successes.” Problems 5.43, 45, 53, and 56.
P(AUB) = P(A) + P(B)
„ This is clear from the Venn diagram
A
B
„ As a special case, since P(A) + P(A′) = 1,
P(A′) = 1 - P(A)
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The General Addition Rule
Conditional Probability
„ For any two events A and B (need not be
„ The probability P(E) is unconditional in the
mutually exclusive),
P(AUB) = P(A) + P(B) - P(A∩B)
„ This is also clear from the Venn diagram
sense that there is no additional information;
if additional information is provided, the
calculated probability of E is conditional.
„ The notation P(A|B) represents the probability
that A will occur given that B has happened.
Here event B is condition which is given.
„ Even though A and B both are in S, for P(A|B)
the sample space is B and S is no longer
relevant in the calculation.
B
A
Conditional Probability
A Special Multiplication Rule: the Case
of Independent Events
„ The Venn diagram illustration of P(A|B):
„ If the occurrence of B has no effect on the
A∩B
A
„ Mathematically,
P( A | B) =
B
P( A ∩ B)
P (B )
„ Examples: Ex. 6.18, 19; Problems 6.51, 52.
likelihood of A’s occurrence, then A and B are
called (statistically) independent. Examples.
„ Mathematically, this is to say
either P(A|B) = P(A) or P(B|A) = P(B)
„ Since P(A|B) = P(A∩B)/P(B), it follows that
P(A∩B) = P(A)⋅P(B)
if A and B are independent.
„ Independence vs. mutually exclusiveness.
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The General Rule of Multiplication
The Three Door Puzzle
„ If A and B are not necessarily independent,
„ Behind the three closed doors, only one
then
P(A∩B) = P(A)⋅P(B|A) = P(B)⋅P(A|B)
„ Clearly, the independent case is a special
case of this general rule.
„ Examples: Examples 6.21 - 25; Problems
6.59 and 6.61.
„ HW: 5.44, 50, 54, 55 and 57; 6.4, 16, 23, 26,
32, 42, 45, 49, 53, 54, 58, 62, 63, and 67.
contains gold. You get to decide which door
to open. Suppose after you have chosen the
door, the host opens an empty Door. Should
you open a different door if you are allowed?
The Three Door Puzzle: the Answer
„ Answer: you should
always choose to open
a different door.
„ The three possibilities
behind the three doors
are
Door 1 Door 2
Door 3
gold
empty
empty
empty
gold
empty
empty
empty
gold
„ For any door you select,
there is 1/3 chance it has
gold -- you win if you
stay; there is 2/3 chance
it is empty -- you win if
you switch.
„ The key point is that the
host opens an empty
door only after your initial
choice, the host’s choice
depends on yours and
the sample space is not
the two remaining doors.
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