Download Indepedent Events and Conditional Probability 1314 H

Copyright © 2013, L. Nelson
Independent
Events
and
Conditional
Probability
• We rarely measure the Relative Frequency or
Probability of an event in isolation. More
often, we are concerned with the likelihood of
a sequence of events, several events
happening at the same time, or the effect of
one event on another.
• It is common in these situations to use a single
letter to represent a particular event and
symbol notation to represent the situation of
interest.
Copyright © 2013, L. Nelson
Considering more than one event
If we are talking about particular events E and F,
then rather that write out the phrases:
• “the relative frequency of E is 50%”
or
• “the probability of F is 5.7%”, we could write:
RF(E)=0.5 or P(F) = 0.057
Note that when using this type of notation, relative
frequencies and probabilities are expressed as
decimal values between 0 and 1, inclusive.
Copyright © 2013, L. Nelson
Notation
Independent Events
Ex:
Let S represent a snow day
Let D represent the last digit on the medallion of
the taxi you’ll take home.
A snow day has almost certainly no effect on whether or not
the taxi medallion will end in a 5, and the taxi you take cannot
change the weather.
These events are independent.
Copyright © 2013, L. Nelson
When the outcome of one event does not
affect the outcome of another, we call them
Dependent Events
Ex:
Let S represent how many hours you study for a test
Let G represent the grade you got on the test
While we may not be able to give a formula that accurately
predicts your grade given x hours of study, studying certainly
affects the likelihood you will remember material.
These events are Dependent
Copyright © 2013, L. Nelson
When the outcome of one event does affect
the outcome of another, we call them
• The affect that dependent events have
on each other may be large or small.
• When two events are dependent, all we
know is that they have some affect on
each other, whether or not we can
quantify it.
Copyright © 2013, L. Nelson
Regarding Dependent Events:
• When considering events in a sequence, the
outcome of one may be related to the outcome
of another.
• For example, consider lung cancer (L) and
smoking (S). If you’re a smoker, it significantly
changes the probability you will develop lung
cancer.
• We would call the probability that you develop
lung cancer given that you are a smoker a
Conditional Probability
= event we want to know the probability of
= condition
Copyright © 2013, L. Nelson
Conditional
Relative Frequency or Probability
• The phrase “given that”, or just “given” is
often used to express conditions.
• A condition may change the relative frequency
or probability of an event, but not necessarily.
• Notation: (continuing the previous example)
P(L S)
Probability of L given S
Copyright © 2013, L. Nelson
Conditional
Relative Frequency or Probability
Ex: Slaps by Stooge – check out the classic Three Stooges here
Source: Two-Way-Stooges-Worksheet-Solution. N.p., n.d. Web. 27 Feb. 2013.
<http://www.docstoc.com/docs/ 101566107/Two-Way-Stooges-Worksheet-Solution>.
Ex:
RF(“Curly did the slapping” | “there were 26 to 30
slaps in the movie”)
The condition tells us to look at data in this row
RF(C|“26-30”) =
4
11
≈ 0.36
Copyright © 2013, L. Nelson
Two-way tables have conditions built into them
Ex: Slaps by Stooge – check out the classic Three Stooges here
Source: Two-Way-Stooges-Worksheet-Solution. N.p., n.d. Web. 27 Feb. 2013.
<http://www.docstoc.com/docs/ 101566107/Two-Way-Stooges-Worksheet-Solution>.
• To calculate conditional probabilities within a twoway table, use the condition to restrict the data you
look at to a single row or column.
• Note that the denominator of your ratio is now the
total of that row (or column), not the absolute total.
Copyright © 2013, L. Nelson
Two-way tables have conditions built into them
You can do the problem set without the following,
but if you’re continuing your study of probability
and statistics after this class, the following slides
are a good introduction to the formal version of
these concepts.
Copyright © 2013, L. Nelson
The rest of the presentation gives the formal,
theoretical version of the topics that have been
covered.
Events A and B are considered
independent if and only if
• P(A|B) = P(A)
and
• P(B|A) = P(B)
This is a formal way of saying
B doesn’t affect A and A doesn’t affect B,
which is how we defined independence earlier
Copyright © 2013, L. Nelson
Formal Definition of
Independent Events
Copyright © 2013, L. Nelson
Independent
Events
and
Conditional
Probability