Probability Rules

1
Lesson 5 - 1

• Probability Rules

2
Objectives

• Understand the rules of probabilities
• Compute and interpret probabilities using the
  empirical method
• Compute and interpret probabilities using the
  classical method
• Use simulation to obtain data based on
  probabilities
• Understand subjective probabilities

3
Vocabulary

• Probability measure of the likelihood of a
  random phenomenon or chance behavior
• Outcome a specific value of an event
• Experiment any process with uncertain results
  that can be repeated
• Sample space collection of all possible
  outcomes
• Event is any collection of outcomes for a
  probability experiment

4
Vocabulary

• Probability model lists the possible outcomes
  of a probability experiment and each outcomes
  probability
• Impossible probability of the occurrence is
  equal to 0
• Certainty probability of the occurrence is
  equal to 1
• Unusual Event an event that has a low
  probability of occurring
• Tree Diagram a list of all possible outcomes
• Subjective Probability probability is obtained
  on the basis of personal judgment

5
The Law of Large Numbers

• As the number of repetitions of a probability
  experiment increases, the proportion with which a
  certain outcome is observed get closer to the
  probability of the outcome.

6
Rules of Probability

• The probability of any event E, P(E), must
  between 0 and 1
• 0 P(E) 1
• The sum of all probabilities of all outcomes,
  Eis, must equal 1
• ? P(Ei) 1
• A more sophisticated concept
• An unusual event is one that has a low
  probability of occurring
• This is not precise how low is low?
• Typically, probabilities of 5 or less are
  considered low events with probabilities of 5
  or lower are considered unusual

7
Empirical Approach

• The probability of an event is approximately the
  number of time event E is observed divided by the
  number of repetitions of the experiment

Frequency of E P(E) relative
frequency of E -------------------------------
--- Total Number of Trials
8
Classical Method

• If an experiment has n equally likely outcomes
  and if the number of ways that an event E can
  occur is m, then the probability of E, P(E), is
• Number of ways that E can occur m
• P(E) -------------------------------------------
  - --------
• Number of possible outcomes n

9
Example 1

• Using a six-sided dice, answer the following
• a) P(rolling a six)
• b) P(rolling an even number)
•  
• b) P(rolling 1 or 2)
• d) P(rolling an odd number)

1/6
3/6 or 1/2
2/6 or 1/3
3/6 or 1/2
10
Example 2

• Identify the problems with each of the following
•  
• a) P(A) .35, P(B) .40, and P(C) .35
• b) P(E) .20, P(F) .50, P(G) .25
•  
• P(A) 1.2, P(B) .20, and P(C) .15
• P(A) .25, P(B) -.20, and P(C) .95

?P gt 1
?P lt 1
P() gt 1
P() lt 0
11
Summary and Homework

• Summary
• Probabilities describe the chances of events
  occurring events consisting of outcomes in a
  sample space
• Probabilities must obey certain rules such as
  always being greater than or equal to 0
• There are various ways to compute probabilities,
  including empirically, using classical methods,
  and by simulations
• Homework
• pg 261-265 9, 11, 12, 15, 18, 25, 26, 32, 34