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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