Download Chapter 4-1

Chapter 4
Probability and Counting Rules
Introduction
“The only two sure things are death and taxes”
A cynical person once said
Introduction
Probability –
The chance of an event occurring.
Introduction
Probability –
The chance of an event occurring.
Probability is the basis for inferential statistics.
Sample Spaces
and Probability
Chapter 4
Section 1
Sample Spaces and Probability
• Flipping a coin, rolling a die, or drawing a card
are all called probability experiments.
Sample Spaces and Probability
• Probability Experiment
– A probability experiment is a chance process that
leads to well-defined results called outcomes.
• Outcome
– An outcome is the result of a single trial of a
probability experiment.
Sample Spaces and Probability
• Probability Experiment
– A probability experiment is a chance process that
leads to well-defined results called outcomes.
• Outcome
– An outcome is the result of a single trial of a
probability experiment.
• A trial means flipping a coin once, roling one die once,
or the like.
Sample Spaces and Probability
• Sample Space
– A sample space is the set of all possible outcomes
of a probability experiment.
• Examples
Sample Spaces and Probability
• Event
– An event consists of a set of outcomes of a
probability experiment.
• Simple Event
– An event with one outcome is called a simple
event.
• Compound Event
– An event with more than one outcome is called a
compound event.
Sample Spaces and Probability
• Tree Diagrams
– A tree diagram is a device of line segments emanating
from a starting point and also from the outcome
point. It is used to determine all possible outcomes of
a probability experiment.
Sample Spaces and Probability
Three basic interpretations of probability
1. Classical Probability
2. Empirical or Relative Frequency Probability
3. Subjective Probability
Classical Probability
• Classical probability uses sample spaces to
determine the numerical probability that an
event will happen.
Classical probability assumes that all outcomes in
the sample space are equally likely to occur.
Classical Probability
• Equally likely events are events that have the
same probability of occurring.
The probability of any event E is
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐸
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
The probability is denoted by
𝑛(𝐸)
𝑃 𝐸 =
𝑛(𝑆)
This probability is called classical probability, and it
uses the sample space S.
Rounding Rule for Probabilities
1. Reduce all fraction to lowest form.
2. Rounded to two or three decimals.
3. If number is really small it is permissible to
round to the first nonzero digit after the
point.
What does “and” and “or” mean
• In probability “and” means at the same time.
• In probability “or” has two cases:
1. Inclusive
2. Exclusive
Probability Rules
1. The probability of any event E is a number
(either a fraction or decimal) between and
including 0 and 1. this is denoted by 0 ≤
𝑃 𝐸 ≤ 1.
2. If an event E cannot occur (i.e. the event
contains no members in the sample space), it
is probability 0.
Probability Rules
3. If an event E is certain, then the probability
of E is 1.
4. The sum of all probabilities of all the
outcomes in the sample space is 1.
Complementary Events
• The complement of an event E is the set of
outcomes in the sample space that are not
included in the outcomes of event E. the
complement of E is denoted by 𝐸( read “E
bar”).
Complementary Events
Rule for complementary events
𝑃 𝐸 = 1 − 𝑃 𝐸 𝑜𝑟
𝑃 𝐸 = 1 − 𝑃 𝐸 𝑜𝑟
𝑃 𝐸 +𝑃 𝐸 =1
Classical Probability
• Venn Diagrams
– One pictorial representation of probabilities.
– Read from your book the two paragraphs on Venn
diagrams (pg. 190-191).
Empirical Probability
• Empirical Probability
– Empirical probability relies on actual experience to
determine the likelihood of outcomes.
– Difference between classical and empirical
probabilities is that classical assumes certain
outcomes are equally likely.
Formula for Empirical Probability
• Given a frequency distribution, the probability
of an event being in a given class is
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠
𝑓
𝑃 𝐸 =
=
𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑛
This probability is called empirical probability and is
based on observation.
Empirical Probabilities
Examples
Empirical Probability
• Empirical probabilities can also be found by
using a relative frequency distribution.
– The frequencies are the same as the relative
frequencies.
Law of Large Numbers
• The Law of large numbers claims that if an
experiment is conducted enough then the
Classical probability will be approximately the
same as the empirical probability.
Subjective Probability
• Subjective Probability
– Subjective probability uses a probability value
based on an educated guess or estimate,
employing opinions and inexact information.
– This guess is based on the person’s experience
and evaluation of a situation.