Transcript AL6-1

Activity 6 - 1
Gerolamo Cardano, Italian
mathematician, wrote the
first book about Probability
in 1560.
Chances Are!
Odds are ….
1 : 576,000
of being hit by lightning
1 : 800,000
of dating a supermodel
1 : 3,000,000
of seeing a UFO
1 : 14,000,000
of winning a scratch-off lottery
Objectives
• Determine relative frequencies for a collection of
data
• Determine both theoretical and experimental
probabilities
• Simulate an experiment and observe the law of large
numbers
• Identify and understand the properties of probability
Vocabulary
• Relative Frequency – what percentage of the whole is some
interested part
• Event – an occurrence of something, like rolling a six on a single die
• Probability of an Event – the chances of an event occurring
• Random –
• Sample Space –
• Probability Distribution –
• Theoretical Probability –
• Experimental Probability –
• Simulation –
Activity
The study of probability began with mathematical
problems arising from games of chance. In 1560, Italian
Gerolamo Cardano wrote a book about games of chance.
This book is considered to be the first written on
probability. Two French mathematicians, Blaise Pascal
and Pierre de Fermat, are credited by many historians with
the founding of probability theory. They exchanged ideas
on probability theory in games of chance and worked
together on the geometry of the die.
Although probability is most often associated with games
of chance, probability is used today in a wide range of
areas, including insurance, opinion polls, elections,
genetics, weather forecasting, medicine, and industrial
quality control.
Activity cont
Complete the following table (which lists years of service
of the 789 workers at High Tech Manufacturing).
Seniority Less than 1 Year 1 – 10 Years More than 10 Years
Male
82
361
47
Female
49
202
48
Total
131
563
95
Total
490
299
789
What is the total number of males?
490
What is the total number of workers with 1-10 years of
service?
563
Activity cont
Convert the table to a table of Relative Frequencies
(Example: 82/789 = .104)
Seniority Less than 1 Year 1 – 10 Years More than 10 Years
Male
0.104
0.458
0.060
Female
0.062
0.256
0.061
Total
0.166
0.714
0.120
Total
0.621
0.379
1.000
If a person is randomly selected from the company, what
is the probability that ….?
a)They are a male
490/789 = 0.621
b)They have less than one year experience
131/789 = 0.166
c)They are a woman with 1-10 years experience
d)They don’t work for the company
202/789 = 0.256
0/789 = 0
Idea of Probability
Chance behavior is unpredictable in the short run, but
has a regular and predictable pattern in the long run
The unpredictability of the short run entices people to
gamble and the regular and predictable pattern in the
long run makes casinos very profitable.
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 frequency.
Relative Frequency
• Relative frequency is the percentage that the
observed makes up of the whole
• Its found by dividing the number of a given
category by the total number of values
• It is equivalent to the Experimental Probability
Probability
• Experimental Probability
– Based on observed frequencies of events
frequency of the event
Probability of an event = ------------------------------------------total number of observations
• Theoretical Probability
– Based on theoretical frequency of events
number of outcomes of the event
Probability of an event = --------------------------------------------------total number of possible outcomes
Laws of Probability
Let P(x) be the probability that event x occurs
• Collection of all possible outcomes is called
the sample space
•
•
•
•
0 ≤ P(x) ≤ 1 for all events x in sample space
Sum of all P(x) for all events x must equal 1
P( certainty ) = 1
P( impossibility ) = 0
Law of Large Numbers
As the number of repetitions of a probability
experiment increases, the proportion
(experimental probability) with which a certain
outcome is observed get closer to the
theoretical probability of the outcome.
Random Selection
Laws of Probability depend on the supposition
that all objects in the collection have an equal
chance of being selected
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
– Law of Large Numbers: as the number of trials is increased the
experimental probability approaches theoretical probability
– Properties of Probability:
a) Sum of probabilities of all possible events must equal one
b) Probability of any single event must be between 0 and 1
c) Probability of impossibility is zero
d) Probability of certainty is one
– Theoretical Probability = (number of outcomes in event) / (total
number of all possible outcomes)
– Experimental Probability = (number of observed occurrences of
event) / (total number of observations)
• Homework
– pg 706 – 712; problems 1, 5, 7, 8, 9, 11