Intro to Probability
The set of all possible outcomes for an experiment is the sample space.
An event is a SUBSET of a SAMPLE SPACE.
If an event has only one possible outcome, it is called a simple event.
If an event is equal to the entire sample space, it is called a certain event.
If an event consists of the empty set, it is called an impossible event.
Because events are sets, we can use set notation and operations with events.
SET OPERATIONS FOR EVENTS
Let E and F be events for a sample space S.
E F means that both events E & F occur.
E F means that event either E or F (or both) occur.
E occurs if E does not occur.
Two events that cannot occur at the same time (such as getting both a head and
a tail on the same coin toss) are called mutually exclusive events.
Mathematically, events E and F are mutually exclusive events if E F = { }.
(By definition, mutually exclusive events are disjoint sets.)
Example
Let S= {1, 2, 3, 4, 5, 6} be the sample space for rolling a single fair die.
Suppose O = {1, 3, 5} is the event that the die roll is odd
and that E = {2, 4, 6} is the event that the roll is even.
Because O E = { }, O and E are mutually exclusive events.
BASIC PROBABILITY PRINCIPLE
Let S be an equally-likely sample space and let the event E be a
subset of S. The probability that event E occurs is
WARNING: Only applies when the outcomes are equally likely.
We can formally define the probability of an outcome in sample space as a number between 0 and 1,
inclusive, such that the sum of all the outcomes in the sample space must be 1 and from that, we can then
define the probability of an event E, written P(E), as the sum of the probabilities of the outcomes that make
up E. However, for the purposes of the probability calculations that we will be doing, the Basic Probability
Principle (above) is the result that you will find to be most practically useful.
OTHER PROPERTIES OF PROBABILITY
Let S be a sample space with n distinct outcomes,
s1, s2, ... , sn and let p1, p2, ... , pn
be the associated probabilities of occurrence.
1. The probability of each outcome is between 0 & 1: 0 < P(E) < 1
(If probability is 0, it is an impossible event; if probability is 1, it is a certain event.)
2. The sum of the probabilities of all possible outcomes is 1.
( p1 + p2 + ... + pn = 1 )
Example
Let S= {1, 2, 3, 4, 5, 6} be the sample space for rolling a single fair die
and O = {1, 3, 5} is the event that the die roll is odd.
What is the probability of the event O (rolling an odd number)?
Example: Suppose you roll 2 dice. What is the probability of a sum of 9?
sample space:
11
21
31
41
51
61
12
22
32
42
52
62
13
23
33
43
53
63
14
24
34
44
54
64
15
25
35
45
55
65
16
26
36
46
56
66
Remember the union rule for sets:
n(E
F) = n(E) + n(F) - n(E
F)
If we divide through by n(S):
n(E F)
n(S)
n(E)
n(S)
=
+
n(F)
n(S)
-
n(E F)
n(S)
we get:
The Union Rule for Probability
P(E
F) = P(E) + P(F) - P(E
if E and F are mutually exclusive events, P(E
F)
F) = { }, giving us
The Union Rule for Probability
for Mutually Exclusive Events
P(E
F) = P(E) + P(F)
Example (when the rule for mutually exclusive events can't be used):
A single ordinary fair die is rolled. What is the probability of getting an even
number or a number greater than 4 ?
'EVEN' = { 2, 4, 6 } '>4' = { 5, 6 }
Note: 'Even' and 'Greater than 4' are mutually NOT exclusive events.
P('Even' OR '>4') = P('Even') + P('>4') - P('Even' AND '>4')
= 3/6 + 2/6 - 1/6 = 4/6 = 2/3
Example (when the rule for mutually exclusive events can be used):
A single ordinary fair die is rolled. What is the probability of getting a 1 or a 5 ?
Note: 'Rolling a 1' and 'Rolling a 5' are mutually exclusive events.
P(1 or 5) = P(1) + P(5) = 1/6 + 1/6 = 2/6 = 1/3
Complement Rule
P(E) = 1 - P(E ) & P(E ) = 1 - P(E)
Some of the problems that we will solve involve a regular deck of playing cards.
If you are not already familiar with this, take a little time to learn.
There are 52 cards divided into 4 suits:
Clubs ( ) & Spades ( ) are black while Hearts ( ) & Diamonds ( ) are red.
There are 4 of each type of card (one of each suit).
Besides number cards (2-10), there are Aces (A), Kings (K), Queens (Q) & Jacks (J).
There are a total of 12 cards with actual faces on them.
These 4 Kings, 4 Queens & 4 Jacks are called FACE CARDS.
EXAMPLE: One card is randomly selected from a standard deck of cards.
What is the probability of drawing a six?
Solution: There are 4 sixes out of 52 cards, so by the Basic Probability Principle:
EXAMPLE: Five cards are drawn from a well-shuffled deck of cards.
What is the probability that all five are clubs?
Solution: First note that when drawing cards, order does not matter...so
counting the cards involves a COMBINATION.
There are C(13,5) ways of drawing 5 cards just from the 13 spades in the deck.
There are C(52,5) ways of drawing 5 cards from any of the 52 cards in the deck.
So, by the Basic Probability Principle: