Section 4.2: Simple Probability, Odds and Sports Wagers
§1 Some Definitions
Sample space – the list of everything that can happen in an experiment
Outcome – each result of the sample space
Event – A subset of the sample space
Theoretical probability- what should happen in an experiment
Empirical probability or Relative frequency- What actually happened in an experiment
§2 Probability
The probability of an event is equal to the # of successes divided by the total number of outcomes in the sample
space.
2
3
4
5
6
7
8
9
10
11
12
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
(3,1)
(3,2)
(3,3)
(3,4)
(4,4)
(5,4)
(6,4)
(4,1)
(4,2)
(4,3)
(5,3)
(6,3)
(5,1)
(5,2)
(6,2)
(6,1)
PRACTICE
1) What is the sample space for rolling a fair die? What is the probability of rolling an even number?
2) What is the sample space of drawing a single card from a standard 52-card deck? What is the probability of
drawing an ace?
3) What is the sample space of rolling two fair die? What is the probability of rolling a seven?
NOTE: An event that can never happen has probability of 0. An event that will definitely happen has probability
of 1. All probabilities are between 0 and 1!
PRACTICE
4) The letters of the word JAMES are written on separate slips of paper and placed in a hat. Two letters are
drawn at random. Use the counting principle to determine the total number of outcomes. List the sample space
of outcomes. Find the probability the first letter is a vowel. Find the probability that both letters are vowels. Find
the probability that neither letter is a vowel.
5) A single card is drawn at random from a standard 52-card deck. Find the probability that the card is a queen.
Find the probability that the card is a king or queen. Find the probability that the card is not a picture card.
Sometimes, we use data in a table to find probabilities. For example, look at the following table.
Iphone
Android
Boys
6
8
Girls
10
5
The values give the total number of responses for boys and girls who have either an Iphone or Android phone.
How many people were surveyed? If one person is chosen at random, what is the probability that the person is a
girl? That the person has an Iphone?
§3 Sports Wagering
This is a common line for sports betting:
Date
01/07/12 101
102
Cincinnati Bengals
Houston Texans
Point Spread
+3 (-110)
-3 (-110)
Total
O 38.5 (-110)
U 38.5 (-110)
Money Line
+165
-190
There are three ways to bet – the point spread, the total number of points scored, and the money line.
Point Spread – The team that has the negative value of the point spread is favored. This means that the sports
books thinks that to keep things even, Houston must win by more than three points. So if you bet on Houston,
you only win your bet if Houston wins by more than three points. By betting on Cincinnati, you would win your
bet if Cincinnati wins the game outright, or loses by less than three points. If Houston wins by exactly three
points, the game is called a push and you simply get your bet back.
The value in parentheses represents how your wager relates to $100. A negative amount represents how much
you would have to bet to win $100. A positive amount represents how much you would win if you bet $100.
Total: This represents the total number of points scored by both teams. You can bet that the total will be over
the given amount (O) or under the given amount (U).
Money Line: This represents a straight up bet - you who think is going to win the game regardless of point
spread. We see that in the column there are positive and negative numbers. Remember, in this game Houston
was the favorite and Cincinnati was the underdog. So if you bet on Houston to win the game straight up, you
would have to bet $190 to win $100. On the other hand, Cincinnati was the underdog. So if you bet $100 on
Cincinnati to win straight up, you would win $165.
PRACTICE
6) Look at the following betting line. This is problems 57-58 in the book.
Date
10/24/10 111
112
Oakland Raiders
Denver Broncos
Point Spread
+8.5 (+105)
-8.5 (-125)
Total
O 43 (-110)
U 43 (-110)
Money Line
+350
-450
The result of the game was Oakland 59, Denver 14.
If you wagered $30 on Oakland with the point spread, what would be your net gain/loss?
If you wagered $10 on Oakland for a money line bet, what would be your net gain/loss?
If you wagered $55 on the over, what would be your net gain/loss?
HW – 1-8, 12-16, 17-21, 27-43, 48-56