Conditional Probability

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Section 2.4

• Conditional Probability

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First Example

• A shopkeeper phones to tell you that she has two
  new baby beagles. You ask her if she has a
  female puppy, and she replies that she does.
  However, on the way to meet her, you change your
  mind and decide that you would rather have a male
  puppy. What is the probability that she has a
  male puppy?

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Simulation

• With a partner, simulate this experiment using
  coins. Record
• the number of times coins are flipped
• the number of times there is at least one head
• of the times there is a head the number of times
  there is also a tail
• Record results of class.
• Estimate the desired probability empirically.

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Beagle Problem

• She has two beagle puppies.
• More info At least one of the puppies is a
  female.
• P(at least one is male at least one is
  female)2/3.
• is read given that.

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Conditional Probability

• Two events A and B. B has non-zero probability.
• Venn diagram intuition of conditional
  probability.
• Definition.

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A Second Example

• Suppose you roll two fair six-sided dice, one of
  which is red and one of which is blue. You
  record the numbers showing on each die. Let A
  the blue die comes up 3 B at least one of
  the dice comes up 3 C the sum of
  the dice is 7
• Find P( A B ), P( B A ), P( A C ), P( C
  A), P( B C ), P( C B ).

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Multiplication Rule

• The following multiplication rule follows
  immediately from the definition of conditional
  probability P( A ? B ) P(B) P( A B )
• A third example. Suppose our experiment is to
  draw 2 cards from a standard deck without
  replacement and record the sequence of cards
  selected (an ordered sample without
  replacement). Let B the first card is a
  Spade A the second card is a Spade
• Find P( A B ) and P( A B? ), and use these to
  compute P( A ? B ) and P( A ? B? ).

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A Fourth Example

• Suppose you know that exactly 4 light bulbs out
  of a box of 10 bulbs are defective. You plan to
  locate the defective bulbs by drawing the bulbs
  out one at a time and testing them (without
  replacement). Find the probability that(a)
  you find the 4th defective bulb on or before the
  seventh test.
• (b) you find the 4th defective bulb on the
  seventh test.

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Working with Conditionals

• For some experiments you can calculate P(A ?
  B) and P(B) . This can be used to find P( A B
  ) . What examples did we do this on?
• Other times you can calculate P(B) and P( A
  B ). This can be used to find P(A ? B) .
  What examples did we do this on?

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Be careful with conditionals.

• Example. Suppose that a person is selected at
  random from a large population of which 1 are
  drug users and that a drug test is administered
  that is 98 reliable. If a person tests positive
  for drugs, what is the probability that they are
  a drug user?
• Notation
• D (drug user) F(tests positive)
• D ? (not a drug user) F ? (tests negative)
• Want
• P(DF)
• Have
• P(D) 0.01 P(D ? )0.99
• P(FD)0.98 P(F ? D)0.02
• P(FD ? )0.02 P(F ? D ? )0.98

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Calculation
Tree Diagram Note. We are omitting
Bayes Theorem, but not example 2.30.
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P-values

• P-values, a step in hypothesis tests, are
  conditional probabilities.
• In the hypothesis test example from Day One, let
• Agroup A gets 3 (what was observed) or fewer
  black cards (black cards represent participants
  who beat threshold)
• B cards dealt randomly (observed difference
  between groups is due to random variation.)
• B? observed difference between groups is due to
  the observer who gets a prize if Group A member
  beats threshold
• P-value is P(AB).
• A common mistake is to interpret the P-value as
  P(B A).