ARE YOU FOR OR AGAINST A MILITARY
STRIKE AGAINST SYRIA?
1.
2.
3.
4.
For
Against
It’s a lot more complicated
Where’s Syria?
17%
17%
17%
17%
17%
17%
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1- 1
1
2
3
4
5
6
UPCOMING IN CLASS

Wednesday’s Quiz 2, covers HW3 of the material
learned in class. (9/11)

Sunday HW4 (9/15)

Part 2 of the data project due next Monday (9/16)
CHAPTER 12
Probability Properties: Addition and
Multiplication under Disjoint and
Independence
DEALING WITH RANDOM PHENOMENA
A random phenomenon is a situation in which we
know what outcomes could happen, but we don’t
know which particular outcome did or will
happen.
 In general, each occasion upon which we observe
a random phenomenon is called a trial.
 At each trial, we note the value of the random
phenomenon, and call it an outcome.
 When we combine outcomes, the resulting
combination is an event.
 The collection of all possible outcomes is called
the sample space.

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1- 4
WHICH OF THE FOLLOWING IS THE SAMPLE
SPACE FOR RECORDING THE ORDER OF HEADS
AND TAILS WHEN TOSSING 3 COINS?
25%
1.
2.
3.
4.
25%
25%
25%
{HHT, HTT, HTH, TTT}
{HHH,HHT,HTH, HTT,
THT,TTH, THH, TTT}
{H,T, HH, HT, TH TT}
{H,T}
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1- 5
1
2
3
4
A FAMILY HAS 5 CHILDREN. WHAT IS THE
SAMPLE SPACE FOR THE NUMBER OF BOYS IN
THE FAMILY?
1.
2.
3.
4.
{1,2,3,4,5}
{0,1,2,3,4,5}
{0,1,2,3,4}
{1,2,3,4}
25%
25%
25%
25%
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1- 6
1
2
3
4
DURING A BASKETBALL GAME A PLAYER TAKES
A SHOT. WHAT IS THE SAMPLE SPACE FOR THE
NUMBER OF POINTS?
1.
2.
3.
4.
{0, 1, 2, 3}
{-1, 1, 2, 3}
{1, 2, 3}
{0, 1, 2}
25%
25%
25%
25%
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1- 7
1
2
3
4
MODELING PROBABILITY WITH RELATIVE
FREQUENCY

The probability of an event is the number of
outcomes in the event divided by the total
number of possible outcomes.
P(A) =
# of outcomes in A
# of possible outcomes
THE LAW OF LARGE NUMBERS (CONT.)
The Law of Large Numbers (LLN) says that the longrun relative frequency of repeated independent events
gets closer and closer to a single value (i.e. true
probability).
 When thinking about what happens with combinations
of outcomes, things are simplified if the individual
trials are independent.

Roughly speaking, this means that the outcome of one trial
doesn’t influence or change the outcome of another.
 For example, coin flips are independent.


Because this definition is based on repeatedly
observing the event’s outcome, this definition of
probability is often called empirical probability.
Slide
1- 9
EVENT SIMULATOR
Coin Toss LLN Experiment (Redline)
 http://socr.ucla.edu/htmls/SOCR_Experiments.ht
ml

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1- 10
FORMAL PROBABILITY
1.
Two requirements for a probability:


A probability is a number between 0 and 1.
For any event A, 0 ≤ P(A) ≤ 1.
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1- 11
FORMAL PROBABILITY (CONT.)
2.
Probability Assignment Rule:


The probability of the set of all possible outcomes of a
trial must be 1.
P(S) = 1 (S represents the set of all possible outcomes.)
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ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
A,B,C,D,E
A,B,C
A,B,D,E
A,B,C,D
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1
2
3
4
THE 68-95-99.7 RULE (CONT.)

The following shows what the 68-95-99.7 Rule
tells us:
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FORMAL PROBABILITY (CONT.)
3.
Complement Rule:


The set of outcomes that are not in the event A is
called the complement of A, denoted AC.
The probability of an event occurring is 1 minus
the probability that it doesn’t occur: P(A) = 1 –
P(AC)
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1- 15
FORMAL PROBABILITY (CONT.)
4.
Addition Rule:

Events that have no outcomes in common (and,
thus, cannot occur together) are called disjoint (or
mutually exclusive).
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1- 16
WHICH OF THE FOLLOWING EVENTS ARE
DISJOINT?
1.
2.
3.
4.
Being black and female
Being republican and
female
Being female and an
athlete
Being female and
having a Y chromosome
25%
25%
25%
25%
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1- 17
1
2
3
4
FORMAL PROBABILITY (CONT.)
4.
Addition Rule (cont.):


For two disjoint events A and B, the probability
that one or the other occurs is the sum of the
probabilities of the two events.
P(A or B) = P(A) + P(B), provided that A and B are
disjoint.
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1- 18
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
=513/1121
=511/1121
=62/1121
=35/1121
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1
2
3
4
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
=513/1121
=511/1121
=(62+35)/1121
=35/62
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1- 20
1
2
3
4
FORMAL PROBABILITY
5.
Multiplication Rule (cont.):


For two independent events A and B, the
probability that both A and B occur is the product
of the probabilities of the two events.
P(A and B) = P(A) x P(B), provided that A and B
are independent.
FORMAL PROBABILITY - NOTATION
Notation alert:
 In this text we use the notation P(A or B) and P(A
and B).
 In other situations, you might see the following:
P(A  B) instead of P(A or B)
 P(A  B) instead of P(A and B)

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1- 22
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
1/3
3
1/6 + 1/6 +1/6
1/6 * 1/6 * 1/6
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1
2
3
4
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
1/2
2
1/2 + 1/2 +1/2
1/2 * 1/2 * 1/2
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1
2
3
4
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
2/6*2/6*2/6
2/6+2/6+2/6
1- 2/6+2/6+2/6
4/6 * 4/6 * 4/6
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1- 25
1
2
3
4
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
1/6*1/6*1/6
1-1/6*1/6*1/6
1-5/6*5/6*5/6
5/6 * 5/6 * 5/6
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1
2
3
4
ENTER QUESTION TEXT...
25%
1.
2.
3.
4.
25%
25%
25%
1/6*1/6*1/6
1-1/6*1/6*1/6
1-5/6*5/6*5/6
5/6 * 5/6 * 5/6
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1
2
3
4
WHAT CAN GO WRONG? (CONT.)

Don’t multiply probabilities of events if they’re
not independent.


The multiplication of probabilities of events that are
not independent is one of the most common errors
people make in dealing with probabilities.
Don’t confuse disjoint and independent—disjoint
events can’t be independent.
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1- 28
CAR PROBLEM


A consumer organization estimates that over a 1year period 16% of cars will need to be repaired
once, 7% will need repairs twice, and 1% will
require three or more repairs.
Suppose you own two cars.
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WHAT IS THE PROBABILITY THAT BOTH
CARS WILL NEED REPAIRED THIS YEAR?
1.
2.
3.
4.
.76
.24
.5776
.0576
25%
25%
25%
25%
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1.
2.
3.
4.
WHAT IS THE PROBABILITY THAT NEITHER
CAR WILL NEED REPAIRED THIS YEAR?
1.
2.
3.
4.
.76
.24
.5776
.0576
25%
25%
25%
25%
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1.
2.
3.
4.
WHAT IS THE PROBABILITY THAT AT LEAST
ONE CAR WILL NEED REPAIRED THIS YEAR?
1.
2.
3.
4.
.76
.24
.5776
.4224
25%
25%
25%
25%
Slide
1- 32
1.
2.
3.
4.
LOTTERY PROBLEM

On September 11, 2002, a particular state
lottery’s daily number came up 9-1-1. Assume
that no more than one digit is used to represent
the first nine months.
Slide
1- 33
WHAT IS THE PROBABILITY THAT THE WINNING
THREE NUMBERS MATCH THE DATE ON ANY
GIVEN DAY THAN CAN BE REPRESENTED BY A
THREE DIGIT NUMBER?
25%
25%
25%
1.
2.
3.
4.
25%
0.000
0.001
0.273
1
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1- 34
1
2.
3.
4
WHAT’S THE PROBABILITY THAT THE WINNING
THREE NUMBER MATCH THE DATE ON ANY
GIVEN DAY THAT CAN BE REPRESENTED BY A 4DIGIT NUMBER?
25%
25%
25%
25%
1.
2.
3.
4.
0.000
0.001
0.273
1
Slide
1- 35
1
2.
3.
4
WHAT’S THE PROBABILITY THAT A WHOLE YEAR
PASSES WITHOUT THE LOTTERY NUMBER
MATCHING THE DAY?
1.
2.
3.
4.
0.001
0.694
0.761
0.999
25%
25%
25%
25%
Slide
1- 36
1.
2.
3.
4.
UPCOMING IN CLASS

Wednesday’s Quiz 2, covers HW3 of the material
learned in class. (9/11)

Sunday HW4 (9/15)

Part 2 of the data project due next Monday (9/16)