Transcript Probability - Courseworks

1
Probability
Class outline

Probability

definitions

Random Error

Normal Distribution

Probability Distributions and Decision Making
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Probability

What is the probability that I will obtain a head
when flipping a nickel?

What is the probability that a student in this class is
female?

Assume a couple has an equal probability of
having a boy or a girl. Which is more likely to
occur with 4 births?

Bbbb

Gggg

bgbg
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Probability

What is the probability that I will obtain a head
when flipping a nickel?


50%
What is the probability that a student in this class is
female?
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Probability

The chance of an event occurring/chance of all
possible events occurring

Ex. 1 Probability of heads on a coin flip =


1(heads)/2(heads or tails)
Ex. 2 Probability of rolling a two on a die

1/6
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Probability

Independent


Mutually exclusive


Not influenced by probability of another event
Cannot occur at the same time
Conditional

Dependent upon another event
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Probability Rules: Mutually
Exclusive Events

Probability of A OR B (heads or tails)

Probability of A + Probability of B


.5 + .5 = 1 or 1-0 = 1
Multiple Events (e.g. Flip a coin twice)

Probability of A AND B (Head on first flip, tails on
second)

Probability of A * Probability of B


.5*.5 = .25
Heads first flip, heads OR tails on second)

Probability of A * (Probability of A or B)

.5*1 = .5
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Probability Rules: Mutually
Exclusive Events

Probability of three heads in a row


.5*.5*.5=.125
Probability of two heads in a row and a head or
tail

.5*.5*(1)=.25
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Practice

What is the probability of obtaining five (5) heads
in a row?

What is the probability of obtaining a head after
obtaining four heads in a row?
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Practice

During World War Two I

Japanese Ace pilot Saburo Sakai in Samurai recalled his
experiences on an airbase in New Guniea: “The enemy did his
best to make sure we did not sleep at night. They almost
unfailingly attacked our air base, dropping bombs in the
darkness.

It got so bad that we often abandoned our sleeping quarters
and went out to the runway after dark to sleep in the craters
dug by enemy bombs. Our theory was that there little
probability of an enemy bomb striking exactly where one had
fallen previously.”1010
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1. What assumption did Saburo Sakai and his colleagues make
about the relationship about where bombs fall?

2. Assume that bombing during World War Two was relatively
inaccurate and within a five mile radius essentially random. If
Saburo and his colleagues wanted the best chance for a
good night’s sleep, where should they sleep, in their beds or in
craters?
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Random Error

Estimate my height


Write on a piece of paper
Guess what the temperature is outside now (don’t
cheat!)
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Random Error

Error

There is a certain amount of error associated with
your estimates

Typically the errors will fall symmetrically around the
true estimate

Normal distribution, a type of probability distribution

Without knowing my true height your best guess
would be the mean

You would be less confident the further away from
the mean a guess is
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Probability Distribution

The frequency of events with a random
component have a distribution

Normal
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Height
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IQ
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T
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Chi-square
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Probability Distribution

Normal Distribution
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In large samples random numbers (e.g. coin flip, roll
of die, prediction errors) will be distributed normally
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Central limit theorem
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Law of large numbers
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Central limit theorem
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When numerous random samples are drawn from
a population distribution, the shape of the
distribution will resemble a normal distribution if
enough samples are drawn (law of large
numbers)
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Probability Distributions
and Decision Making

An example with a coin
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One of the coins in my hand is “fake” or has two
heads

How many heads do I have to obtain before you
are willing to bet the coin is fake?
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Probability Distributions
and Decision Making

An example with a coin

One of the coins in my hand is “fake” or has two
heads

How many heads do I have to obtain before you
are willing to bet the coin is fake?

Probability of one flip without a heads 50%

Probability of two flips without a heads 25%
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Probability of three flips without a heads 12.5%
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Probability of four flips without a heads 6.25%
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Probability Distributions
and Decision Making

An example with a die

One of the dies in my hand does not have a one

How many rolls do you have to observe before
determining if the die is fake?
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Probability Distributions
and Decision Making

An example with a die

One of the dies in my hand does not have a one

How many rolls do you have to observe before
determining if the die is fake?
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Probability of one roll without a one 83.33%
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Probability of two rolls without a one 69.4%
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
.

.

.
Probability of ten rolls without a one 16.2%

.
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Probability Distributions
and Decision Making

If we know the probability distribution (e.g. how
likely to get x number of heads or x number of rolls
without a one) we can make informed decisions
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Probability Distributions
and Decision Making

The link to Social Science and Planning

If we know the properties of a given probability
distribution we can estimate the probability of an
event occurring

Ex. Normal distribution

The probability of x depends on the mean and the
standard deviation

Z-score tells us how many standard deviations x is from
the mean

We can use the z-score to determine the probability of
x
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Z STATISTIC
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𝑍=

𝑥−𝜇
𝜎
Where
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x is the observed item
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𝜇 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛

σ is the standard deviation
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Example in class problems

Vince Yosarian works for the Bureau of Forms. His
manager has reprimanded Vince because she
feels that he has not performed up to standard for
processing forms at the bureau, the crucial part of
the job. At the bureau, the number of forms
processed by employees is normally distributed,
with a mean of 67 forms per employee per day,
with a standard deviation of 7. His manager has
calculated that Vince’s average rate is 50 forms
processed per day. What percentage of
employees at the bureau process fewer forms
than Vince? What percentage of employees at
the bureau process more forms than Vince? Does
the manager’s complaint seem justified or not?
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Example in class problems
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1. Calculate z-statistic
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z = (50-67) /7 = -2.43
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Example in class problems
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1. z = (50-67) /7 = -2.43
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Associated probability = .4925
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Interpretation: 99.25% of workers are faster
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Example
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Parnelli Jones, a vehicle manager for the northeast region
of the forestry service, is charged with purchasing
automobiles for service use. Because forestry service
employees often have to drive long distances in isolated
regions, Parnelli is very concerned about gasoline mileage
for the vehicle fleet. One automobile manufac-turer has
told him that the particular model of vehicle that interests
him has averaged 27.3 miles per gallon in road tests with a
standard deviation of 3.1 (normal distribution). Parnelli
would like to be able to tell his superiors at the forestry
service that the cars will get at least 25 miles to the gallon.
According to the road test data, what percentage of the
cars can be expected to meet this criterion?

Parnelli also thinks that his superiors might settle for cars
that got 24 miles to the gallon or better. What percentage
of the cars can be expected to meet this criterion?
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Example in class problems

2.
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a. z = (25-27.3) /3.1 = -.7
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
Associated probability = .24196
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Interpretation: 24.2% of cars will not meet criterion

Please note that the z-table in the textbook gives the
percentage between the mean and .7 standard deviations
below the mean. 24.2% represents the number that falls below
that value
b. z = (24-27.3) /3.1 = -1.1
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Associated probability .13567
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Interpretation: 13.6% of cars will not meet criterion

Please note that the z-table in the textbook gives the
percentage between the mean and .7 standard deviations
below the mean. 13.6% represents the number that falls below
that value
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T-distribution

T-statistic
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Working with small samples
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Underlying population is normal
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𝑡=
𝑥−𝑋
𝑠.𝑒.
with n—1 df

x is the observed item

𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛

s.e. is the standard deviation of sampling means or the
standard error

N sample size

Df is degrees of freedom
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