Transcript Section 1

Lesson 5 - 1
Probability Rules
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
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
Vocabulary
• Probability model – lists the possible outcomes of a
probability experiment and each outcome’s
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
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.
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, Ei’s,
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
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
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
Example 1
Using a six-sided dice, answer the following:
a) P(rolling a six)
1/6
b) P(rolling an even number)
3/6 or 1/2
b) P(rolling 1 or 2)
2/6 or 1/3
d) P(rolling an odd number)
3/6 or 1/2
Example 2
Identify the problems with each of the following
a) P(A) = .35, P(B) = .40, and P(C) = .35
∑P > 1
b) P(E) = .20, P(F) = .50, P(G) = .25
∑P < 1
c) P(A) = 1.2, P(B) = .20, and P(C) = .15
P() > 1
d) P(A) = .25, P(B) = -.20, and P(C) = .95
P() < 0
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