Ch 15: Probability Rules Recall That

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Ch 15 Probability Rules! Recall That

• For any random phenomenon, each trial generates
  an outcome.
• An event is any set or collection of outcomes.
• The collection of all possible outcomes is called
  the sample space, denoted S.

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Events

• Also recall
• When outcomes are equally likely, probabilities
  for events are easy to find just by counting.
• When the k possible outcomes are equally likely,
  each has a probability of 1/k.
• For any event A that is made up of equally likely
  outcomes,

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Not all outcomes are equally likely!

• 1 Determine the sample space (defined as the
  list of all possible outcomes) for the following
  examples. Are all outcomes equally likely? For
  ones that arent show how!
• Flipping a quarter and a dime and noting whether
  each lands heads or tails.
• The number of boys in a family with 3 children
• Flipping a coin until you get a head or 3 tails
  in a row
• Tossing 2 dice and recording the larger of the 2.
• Hint- when figuring out if you have the sample
  space represented, sum up all the probabilities
  and see if they come to 1. If they dont, youve
  miscalculated or forgotten an outcome!

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The General Addition Rule

• When two events A and B are disjoint, we can use
  the addition rule for disjoint events from
  Chapter 14
• P(A or B) P(A) P(B)
• However, when our events are not disjoint, this
  earlier addition rule will double count the
  probability of both A and B occurring. Thus, we
  need the General Addition Rule.
• But before working with this, lets remember the
  first 3 rules of data analysis (does anyone?),
  because they are the same for working with
  probability rules!

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The First Three Rules for Working with
Probability Rules

• Make a picture.
• Make a picture.
• Make a picture.
• The most common kind of picture to make for
  probabilities is called a Venn diagram.

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The General Addition Rule - Illustrated

• General Addition Rule
• For any two events A and B,
• P(A or B) P(A) P(B) P(A and B)
• The following Venn diagram shows a situation in
  which we would use the general addition rule

Note sometimes people try to size the circles
according to how probable the event is. Its nice
if you can attempt to do this roughly, but dont
worry about it too much.
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Example

• For a standard 6-sided dice what is the
  probability that the face will be
• a multiple of 2?
• a multiple of 3?
• Either a multiple of 2 or 3?

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It Depends

• Back in Chapter 3, we looked at contingency
  tables and talked about conditional
  distributions.
• When we want the probability of an event from a
  conditional distribution, we write P(BA) and
  pronounce it the probability of B given A.
• A probability that takes into account a given
  condition is called a conditional probability.

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It Depends The Formula!

• To find the probability of the event B given the
  event A, we restrict our attention to the
  outcomes in A. We then find the fraction of those
  outcomes B that also occurred.
• Note P(A) cannot equal 0, since we know that A
  has occurred.

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Example Conditional Probabilities

• 478 children were surveyed to pick their top
  priority of 3 goals. Heres the contingency table
  (remember Ch 3?).
• Whats the probability a kid from this group is a
  girl who wants to excel at sports?
• Whats the probability the kid from this group is
  a girl?
• Whats the probability that a girls first
  priority is to excel at sports?
• This is just a convenient rephrasing of Whats
  the probability a kids first priority is to
  excel at sports, given that kid is a girl?

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The General Multiplication Rule

• When two events A and B are independent, we can
  use the multiplication rule for independent
  events from Chapter 14
• P(A and B) P(A) x P(B)
• However, if events are not independent, this
  earlier multiplication rule does not work. We
  need the General Multiplication Rule.
• We have encountered this rule in the form of
  conditional probabilities. Rearranging the
  equation that defines conditional probability
  gives the General Multiplication Rule
• For any two events A and B,
• P(A and B) P(A) x P(BA)
• or
• P(A and B) P(B) x P(AB)

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Example Why We Need the General Multiplication
Rule

• In the prior survey, we saw that there are 30
  girls from the group who wished to excel at
  sports.
• Thus we calculate P(Girl AND Sports) 30/478
  .063.
• If a kids priorities were independent of gender,
  then wed be able to calculate the probability
  that a kid is a girl who wants to excel at sports
  by doing the following
• The probability a kids first priority is to
  excel at sports 90/478 .188
• The probability the kid is a girl 251/472
  .525
• But .188.525 .099 which is NOT .063
• This calculation illustrates the following

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Independence

• Independence of two events means that the outcome
  of one event does not influence the probability
  of the other.
• With our new notation for conditional
  probabilities, we can now formalize this
  definition
• Events A and B are independent whenever P(BA)
  P(B). (Equivalently, events A and B are
  independent whenever P(AB) P(A).)

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Independent ? Disjoint

• Remember from Ch 14, Disjoint events cannot be
  independent! Well, why not?
• Since we know that disjoint events have no
  outcomes in common, knowing that one occurred
  means the other didnt.
• Thus, the probability of the second occurring
  changed based on our knowledge that the first
  occurred.
• It follows, then, that the two events are not
  independent.
• A common error is to treat disjoint events as if
  they were independent, and apply the
  Multiplication Rule for independent eventsdont
  make that mistake.

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Drawing Without Replacement

• Sampling without replacement means that once one
  individual is drawn it doesnt go back into the
  pool.
• We often sample without replacement, which
  doesnt matter too much when we are taking small
  samples from a large population. (San Francisco
  residents, SFSU students, U.S. voters.)
• However, when drawing from a small population, we
  need to take note and adjust probabilities
  accordingly.
• Example- Problem 36 in Chapter 14. The
  probability of the 11th card from a deck being
  red after weve drawn 10 reds in a row is not
  50, because more black cards remain in the deck
  than red!
• Drawing without replacement is just another
  instance of working with conditional
  probabilities.

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Example of Drawing Without Replacement

• 19. You have a junk box which contains a dozen
  old AA batteries, 5 of which are dead. You
  start picking batteries one at a time and testing
  them.
• You will not put batteries, dead or live, back in
  the drawer.
• Find the probability that
• The first 2 you choose are both good.
• At least 1 of the first 3 works.
• The first 4 you pick all work
• You have to pick 5 batteries in order to find 1
  that works (i.e. you would stop looking as soon
  as you found one)

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Tree Diagrams

• A tree diagram helps us think through conditional
  probabilities by showing sequences of events as
  paths that look like branches of a tree.
• We can use tree diagrams in place of doing lots
  of General Multiplication calculations
• Making a tree diagram for situations with
  conditional probabilities is consistent with our
  make a picture mantra.

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Example Tree Diagrams

• According to a public health study
• 44 of college students engage in binge drinking
  (sequential consumption of 4 to 5 drinks) 37
  drink moderately and 19 abstain.
• Another study shows that 17 of young binge
  drinkers have had an alcohol related accident
  while only 9 of non-bingers have.
• Whats the probability that a randomly selected
  college student will be a binge drinker who has
  had an alcohol related accident?
• We will draw a tree diagram to solve this

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Tree Diagram Example

• First we draw the structure of the tree, starting
  with what type of drinker the student is.
• Then we add their chance of having an accident.
• Multiplying the probabilities gives us the joint
  probabilities.
• What is the probability that any student you meet
  is a binge drinker who had an accident?

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Reversing the Conditioning

• Reversing the conditioning of two events is
  rarely intuitive.
• Suppose we want to know P(AB), but we know only
  P(A), P(B), and P(BA).
• We also know P(A and B), since
• P(A and B) P(A) x P(BA)
• From this information, we can find P(AB)

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Reversing the Conditioning Example

• When we reverse the probability from the
  conditional probability that youre originally
  given, you are actually using Bayess Rule.
• We could write the equation out, but its easier
  to use tree diagrams.
• Example What if we want to find the chance that
  a student who had an alcohol-related accident is
  a binge drinker?
• Is it the same as the chance that a binge drinker
  has an alcohol-related accident?

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What Can Go Wrong?

• Dont use a simple probability rule where a
  general rule is appropriate
• Dont assume that two events are independent or
  disjoint without checking that they are.
• Dont find probabilities for samples drawn
  without replacement as if they had been drawn
  with replacement.
• Dont reverse conditioning naively.
• Dont confuse disjoint with independent.

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What have we learned?

• The probability rules from Chapter 14 only work
  in special caseswhen events are disjoint or
  independent.
• We can use Venn diagrams, tables, and tree
  diagrams to help organize our thinking about
  probabilities.
• We now know the General Addition Rule and General
  Multiplication Rule.
• We also know about conditional probabilities and
  that reversing the conditioning can give
  surprising results.
• We now know more about independence
• Understanding of independence will be important
  throughout the rest of this course.

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More Problems

• 22 56 of all US workers have a retirement
  plan, 68 have health insurance and 49 have both
  benefits. If we select a worker at random
• What is the probability he has neither health
  insurance nor a retirement plan?
• Whats the probability he has health insurance if
  he has a retirement plan?
• Are having health insurance and a retirement plan
  independent events? Explain!
• Are having health insurance and a retirement plan
  disjoint? Explain!
• Hint its useful to make a picture (what type?)
  when we are first working with this problem.

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More Answers

• Constructing a Venn Diagram makes the rest of the
  problem easier to understand.

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Problems

• 45 Dans Diner employees 3 dishwashers. Al
  washes 40 of the dishes and breaks 1 of them,
  while Betty and Chuck each wash 30, but Betty
  breaks only 1 while Chuck breaks 3.
• If you hear a dish break at Dans, whats the
  probability that Chuck was washing it?
• Whats the probability that any dish breaks?
• Do you think its easier to draw the tree or use
  the formula associated with Bayes Rule?

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Answer
And any dish has a chance of breaking of
.004.003.009 .016
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Old Exam Question

• A close friend of yours did not use birth control
  last month and fears she could be pregnant. She
  buys Preggers, a new pregnancy home testing kit.
  There are only 2 outcomes for this test Yup
  (indicating the subject is pregnant) and Nope
  (indicating the subject is not pregnant.)
  Preggers will give a Yup on 95 of the
  pregnant women tested, but is less reliable in
  the other direction, showing Nope only 80 of
  the time when a woman is actually not pregnant.
  Medical data shows that 35 of sexually active
  women of your friends age and health are likely
  to get pregnant if they dont use birth control
  that month.
• What is the chance that she gets a Yup on the
  test?
• Assume your friend takes the test and gets a
  Yup. What is the chance she is actually
  pregnant?
• Preggers offers a money-back guarantee for tests
  that were wrong. Assuming that all of the
  consumers who bought Preggers have the same
  chance of being pregnant as your friend (and that
  everyone will seek the refund if she gets a wrong
  result), calculate the percentage of kit sales
  that Preggers will have to offer refunds for.