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Probability Models
Section 6.2
The Language of Probability
What is random?
Empirical means that it is based on observation rather than theorizing.
Probability describes what happens in MANY trials.
Example 6.9: Long-term relative frequency
Randomness and Probability We call a phenomenon random if
individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency.
Assignment
Page 410 exercises 6.21 – 6.28
Toss a coin…
We cannot know the outcome in advance.
The outcome will be either heads or tails.
Each of these outcomes has the probability of ½.
The basis of all probability models is a list of all possible outcomes and a probability for each outcome.
Sample Spaces
The sample space S of a random phenomenon is the set of all possible outcomes.
To specify S, we must state what constitutes an individual outcome and then state which outcomes can occur.
How to count!
Being able to properly enumerate the outcomes in a sample space will be critical to determining probabilities.
Two techniques are very helpful in making sure you don’t accidentally overlook any outcomes.
These techniques are the tree diagram and the multiplication principle.
Tree Diagram
Toss
a coin
H
T
Roll a
die
1
2
3
4
5
6
1
2
3
4
5
6
H1
H2
H3
H4
H5
H6
T1
T2
T3
T4
T5
T6
Multiplication Principle
If you can do one task in n number of ways and a second task in m number of ways, then both tasks can be done in nXm number of ways.
Nondiscrete sample space
Some sample spaces are simply too large to allow all of the possible outcomes to be listed.
Generate a random decimal number between 0 and 1.
– Nondiscrete sample spaces
With and Without Replacement
Sampling with replacement means that once you’ve made your first selection, you return it so that it can be chosen again.
Sampling without replacement means that you do not return your first selection.
Assignment
Page 416, problems 6.29 – 6.36
Probability of an Event
The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes.
P(A) = The Number Of Ways Event A Can Occur
The Total Number Of Possible Outcomes
A pair of dice is rolled, one black and one white. Find the probability of each of the following events.
1. The total is 10.
2. The total is at least 10.
3. The total is less than 10.
4. The total is at most 10.
5. The total is 7.
6. The total is 2.
7. The total is 13.
8. The numbers are 2 and 5.
9. The black die has 2 and the white die has 5.
10.The black die has 2 or the white die has 5.
112
16
56
1112
16
136
0
118
136
13
Probability Rules All probabilities are between 0 and 1 inclusive
The sum of all the probabilities in the sample space is 1
The probability of an event which cannot occur is 0.
The probability of any event which is not in the sample space is zero.
The probability of an event which must occur is 1.
The probability of an event not occurring is one minus the probability of it occurring.
P(E') = 1 - P(E)
0 ( ) 1P E
The Addition Rule
Two events are disjoint (mutually exclusive) if they have no outcomes in common.
If two events are disjoint, the number of ways one or the other can occur is
( or ) ( ) ( )n A B n A n B
Set Notation
Union
Empty Event
Intersect
Examples
6.13 Complement Rule
6.14 Applying Probability Rules
6.15 Applying Probability Rules
6.16 Applying Probability Rules
Assignment
Page 423, problems 6.37 – 6.44
Independence and the multiplication rule
Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs.
If A and B are independent,
( and ) ( ) ( )n A B n A n B
Independent or not independent?
Example 6.17
Example 6.18
Applying the Multiplication Rule
Example 6.19
Independence and the Complement Rule
Example 6.21
Assignment
Page 430, exercises 6.45 – 6.52
Section Exercises
Page 432, exercises 6.53 – 6.63
