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Chapter 3

# Chapter 3 notes

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McGill University

Mathematics & Statistics (Sci)

MATH 203

Patrick Reynolds

Fall

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Math 203: Chapter 3
Sample Points
Experiment – the process of observation that leads to a single outcome that cannot be
predicted with certainty
Sample point – basic outcome of an experiment
Sample space – collection of all sample points of an experiment
The probability of a sample point is a # between 0 and 1 which measures the likelihood that
the outcome will occur.
Example:
Consider the experiment of tossing 2 coins. One sample point, for example, is (“Observe a
head” and “Observe a tail”)
Tree diagram:
Assigning Probabilities
The probability of a sample point could be taken to be the relative frequency of the occurrence
of the sample point in many repetitions of the experiment. (theoretical)
Of course, this may not be possible in progress.
Subjective probabilities might be assigned based on general information of expert analysis of
the experiment (e.g. weather).
Events
Let pirepresent the probability of sample point i. All sample point probabilities must lie
between 0 and 1, 0 ≤ p i 1. The sum of probabilities of all sample points in a sample space must
be 1: ∑
An event is a collection of sample points. To calculate the probability of an event, sum the
probabilities of the sample points in the event.
(back to example)
Let’s suppose the coins are not balanced. The following probabilities are assigned to each of the
sample points:
HH |
HT |
TH |
TT | Consider the event: “Observe exactly one head” (denoted by A)
“A” consists of the sample points: HT and TH. The probability that A occurs is:
P(HT)+ P(TH) =
Consider the event: “Observe at least one head” (denoted by B)
The event B consists of the sample points: B = {HH, HT, TH}. The probability B occurs is:
P(HH) + P(HT) + P(TH) =
Or
( )
Combinations Rule
Suppose a set of n elements is to be drawn without replacement from a set of N elements. The
number of different possible samples is denote( ) and is equal to
( ) ( )
Where:
( )( ) ( )( )( )
**Note: 0! Is defined to be 1
Ex: supposed you want to select 2 people from a group of 4. There are 6 different ways to do
so.
( ) – pronounced “N choose k”
( )
( ) ( )
Ex: lottery – pick 6 numbers without replacement from 53. You can do this( ) ways
R: > choose(53,6)
[1] 22957480
Unions, Intersections, Complements
Consider event A and event B
Union: the event that occurs if either A or B (or both) occurs. Denoted by (AUB).
- Venn diagram
** event – a collection of sample points
- Union is all sample points in A, B, or both Intersection: the event that occurs if both A and B occur. Denoted by (A∩B)
Example:
Consider the experiment of rolling a fair die. Let A be the event “Roll and even number”. Let B
be the event “Roll a number ≤3”.
A={2, 4, 6}
B = {1, 2, 3}
C
A C{1, 3, 5}
B ={4, 5, 6}
AUB= {1, 2, 3, 4, 5, 6}
(A∩B)C= {2}
AUB ={5}
(A∩B) ={1, 3, 4, 5, 6}
So each sample point in the sample space, S={1, 2, 3, 4, 5, 6} is equally likely, with probability
P(AUB)=
P(A∩B)=
**Complement of A: the event that A does not occur. Denoted by A C
Rule of CoCplements:
P(A) + P(A ) = 1
(Equivalently) P(A) = 1- P(A )
Additive Rule:
( ) ( ) ( ) ( )
Mutually Exclusive Events:
P( )
(equivalently) ( ) ( ) ( )
- Mutually exclusive events are 2 events that cannot happen at the same time (i.e.
flipping a coin – the outcome can either be heads OR tails, NOT both)
Example:
Consider tossing 10 balanced coins. Let A be the event:
A: “Observe at least 1 head”
First, understand the sample space
Basic sample point (example:
HTHHTTHHHT, TTTTTTHHTH Here’s how we might count all the sample points:
2 2 2 2 2 2 2 2 2 2 = 2 = 1024
10
So thereCare 2 = 1024 sample points all equally likely.
Note, A ={T T T T T T T T T T}
C
And P(A )=
P(A) = 1-P(A )
=
=
Additive Rule (ex:)
Reconsider the die experiment:
A={2, 4, 6}
B = {1, 2, 3}
P(A)=
P(B) =
A B={1, 2, 3, 4, 6}
A⋂B={2}
P(A B) = P(A) +P(B) – P(A⋂B)
=
Mutually Exclusive Events
- No intersection
- P(A⋂B)=0
Conditional Probability (3.5 textbook)
Reconsider rolling a fair die
A = {roll even number}
B = {roll a number ≤3}
P(A)=
But suppose we know that event B occurred. What’s the probability of A, given that B

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