Sec. 5.1 & 5.2: Idea of Probability and Probability Models
Basic Vocabulary:
Law of Large Numbers: the proportion of times that a pa4rticular outcome occurs in many repetition s will
approach a single number.
Probability: Long-run relative frequency of a chance outcome.
 Always a number between 0 and 1
 Probability of 0 means it never occurs
 Probability of 1 means it always occurs
**Long series of repetitions  Trials must be INDEPENDENT!
Ways to determine probability:
1. Empirical study actually observe outcomes
2. Compute theoretical probabilities (Sec. 5.2)
3. Educated guess
Random: individual outcomes are uncertain, but there is a regular pattern (distribution) in a large number of
repetitions.
- Flipping (a fair) coin
- Random Number Tables
- Rolling dice
- Drawing from a hat
Discuss Example on p. 295, Aren’t We Due for a Boy?
Sample Space, S: the set of all possible outcomes
Outcome: a single possible result
Probability Model: a description of some chance process consisting of 2 parts: a sample space and a
probability for each outcome.
Event: a collection of outcomes. Subset of the Sample space. Designated by capital letters.
Example: Craps (rolling 2 dice): How many ways can we roll two dice?
S=
11
12
13
14
15
16
21
22
23
24
25
26
31
32
33
34
35
36
41
42
43
44
45
46
51
52
53
54
55
56
61
62
63
64
65
66
36 possible outcomes (first die, second die)
**Event A: “Roll a 5” (sum of 5 in Craps)  A = {1 4, 2 3, 3 2, 4 1}  4 out of 36 So P(A) = 4/36 = 1/9
Gamblers only care about number of dots showing. Sample space for this:
(Remember we’re talking about the SUM of the two dice)
S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
1+1
6+6
Multiplication principle: If you can do one task m number of ways and the other task n number of ways, both
can be done m x n numbers of ways.
Example: Craps
Example: Flipping Coin, Tossing Die
2 dice
First die
1, 2, 3, 4, 5, 6
Second die
1, 2, 3, 4, 5, 6
6
6
6 x 6 = 36 outcomes
Coin
H, T
Die
1, 2, 3, 4, 5, 6
2
6
2 x 6 = 12 outcomes
Example: Flipping 4 coins
2 x 2 x 2 x 2 = 16 outcomes
*You can define your sample space differently depending on the problem.
Ex. When flipping 4 coins, how many heads?
0 Heads
TTTT
1 Head
HTTT
THTT
TTHT
TTTH
2 Heads
HHTT
HTHT
HTTH
THHT
THTH
TTHH
HW: p. 300 # 1, 3, 9, 11 & p. 314 # 39, 41
S = {0, 1, 2, 3, 4}
3 Heads
HHHT
HHTH
HTHH
THHH
4 Heads
HHHH