Math 3680

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Math 3680 Lecture 3 Probability
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• We hear about chance in several different
  contexts
• 1. A certain horse is given 51 odds to win a
  race.
• 2. A banker recommends a certain stock for
  investment.
• 3. Josh Hamilton is batting 0.361 for the season.
• 4. In a recent poll, 48 of Americans approve of
  the presidents performance.
• 5. The chance of being dealt the queen of spades
  is 1 in 52.
• 6. The chance of winning a Pick 4 lottery game is
  1 in 10,000.

We have three different types of
probabilities   1) Expert opinion
(subjective) 2) Relative frequency
(empirical) 3) Equally likely outcomes
(theoretical)
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• Definition The probability of event A is
  denoted P(A).
• 1) Chances are between 0 and 100. That is
• 0 P(A) 1.
• We denote an impossible event (the empty set) by
  Ø
• P(Ø) 0.

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There is a technical difference between the
phrases impossible and probability zero.
Example I pick a real number in the interval
0,1 and you try to guess my number. Its not
impossible, but the probability that you pick my
number is zero. Question Would the same be
true if I told you that I rounded my number to
six decimal places?
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• 2) The chance of something not happening is 100
  minus the chance of it happening. That is,
• P(not E) P(E )
  1 - P(E).
• This event, not E, is referred to as the
  complement of E.

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• 3) When you draw at random, we assume that all
  the possible outcomes are equally likely, so
  that
• number of favorable
  outcomes
• P(E)
  
• number of possible
  outcomes
• The word draw is used to suggest select, or
  pick, or choose. There are two ways that
  draws can be made
• 1. Draws with replacement, such as rolling a die
  repeatedly or tossing a coin repeatedly.  
• 2. Draws without replacement, such as dealing
  cards from a deck or 5 CD random shuffle.

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• 4) If E implies F, then F is more likely
  than E.
• If , then
  P(E) P(F).
• Example Let E a dealt card is a heart
• Let F a dealt card is red

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• Definition Conditional probability.
• The conditional probability that A happens given
  thatB happens is denoted by
• P(A B)
• Formula
• Here, A n B denotes the intersection of A and B
  that is, the event that A and B both happen.

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• Ex Two tickets are drawn without replacement
  from the box
• A) What is the chance that the second ticket is
  ,
• given the first is ? (Conditional
  probability)
•  
• B) What is the chance that the first ticket is
  ?
• (Unconditional probability)
• C) What is the chance that the second ticket is
  ?
• (Again unconditional probability!)
• D) Repeat if the draws are made with replacement.

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• Example Five cards are dealt off the top of a
  well-shuffled deck.
• A) Find the chance that the fifth card is the
  queen of spades.
• B) Find the chance that the fifth card is the
  queen of spades, given that the first four cards
  are hearts.

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• Example A penny is tossed five times.
• A) Find the chance that the fifth toss is a
  head. 
• B) Find the chance that the fifth toss is a
  head, given that the first four tosses are tails.

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Example A box has three tickets red, white
and blue. Two tickets are drawn at random
without replacement. What is the chance that the
red ticket is drawn first, and the white ticket
second?

• Solution 1 Counting the possibilities.
•  
• RW WR BR
• RB WB BW
•  
• So the chance is

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• Solution 2 Multiplication Rule. Imagine a
  large group of people attempting this experiment.
  Roughly one-third will get the first draw
  correct.
• Of these people, about half will get the second
  draw correct. Thus, from the original
  population,
• will get both draws correct.

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• Definition Multiplication Rule.
• The chance that events A and B both happen is
  found by multiplying the chance that event A
  happens by the chance that B happens, given that
  A has happened
• P(A n B) P(A) P(B A)

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• Example A deck of cards is shuffled and two
  cards are dealt. What is the probability that
  both are hearts?
• Method 1 List the possible outcomes.

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Example A deck of cards is shuffled and two
cards are dealt. What is the probability that
both are hearts? Method 2 Use the
multiplication rule.
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• Example Five cards are dealt from a
  well-shuffled deck. What is the probability that
  all 5 are hearts?

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• Example In poker, what is the probability of
  being dealt a flush?

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• Example A die is rolled four times. What is
  the probability that all four rolls are
  different?

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• Definition Independent. We say that two
  events A and B are independent if the outcome of
  event A does not influence the outcome of event
  B. In mathematical notation,
• P(B A) P(B)

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• Example A coin is tossed twice. Let
• A the first flip lands heads
• B the second flip lands heads
• Are events A and B independent?

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• Example A die is rolled once. Let
• A the die lands on four
• B the die lands on six
• Are events A and B independent?

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• Example Two cards are dealt. Let
• A the first card is an ace
• B the second card is an ace
• Are events A and B independent?

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• Example One card is dealt. Let
• A the card is a jack
• B the card is a heart
• Are events A and B independent?

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• Rule of thumb
• When drawing with replacement, the draws are
  independent.
• When drawing without replacement, the draws are
  dependent.

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• Example A die is rolled 10 times. Find the
  chance of ...
• A) getting 10 sixes
•  
• B) not getting 10 sixes
• C) all the rolls showing 5 or fewer dots
• D) getting at least one six
• Are B) and C) the same?

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• Definition Mutually Exclusive. Two events
  are called mutually exclusive (or disjoint) if
  they cannot occur simultaneously.
• WARNING Disjoint is a different concept than
  independent.
• If E and F are disjoint, then P(F E) 0.
• If E and F are independent, then P(F E)
  P(F).

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• Example A coin is flipped twice. Let
• A the first flip lands heads
• B the second flip lands heads
• Are events A and B disjoint?

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Example A die is rolled once. Let A the
die lands on four B the die lands on six
Are events A and B disjoint?
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• Example Two cards are dealt. Let
• A the first card is an ace
• B the second card is an ace
• Are events A and B disjoint?

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• Example One card is dealt. Let
• A the card is a jack
• B the card is a heart
• Are events A and B disjoint?

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• Property If A and B are disjoint, then

P(B)
P(A)
P(A or B)
Note The word or has a different meaning in
probability than in ordinary English usage. It
does not mean exclusive or. That is, A or
B means A, or B, or both A and B.
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• Example A card is dealt from a shuffled deck.
  What is the probability that the card is a face
  card?

Solution P(jack or queen or king) P(jack)
P(queen) P(king)
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• Example A card is dealt from a well-shuffled
  deck. What is the probability that it is a red?
•  
• Solution
• P(heart or diamond) P(heart) P(diamond)
• (mutually exclusive)

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• Example Two dice are rolled. Whats the
  probability that an ace (one dot) appears?
•  
• Common Mistake
• P(ace on first or ace on second)
• P(ace on first ) P(ace on second)

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• Correct Answer 1 Look at a chart of the 36
  possible outcomes for the roll of two dice. There
  are 11 pairs in which an ace appears. So the
  answer is .

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• Because these two events are not mutually
  exclusive, the first (incorrect) method produced
  an answer that was too big instead of
  .
• Subtracting something makes sense. After all,
  the probability of an ace appearing on 7 dice
  cannot be .

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K
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• Correct Answer 2 Use the general formula for
  events which are not mutually exclusive

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P(A n B )
P ( B)
P(A)
P(A or B)
Notice that the Rule of Addition saves a lot of
time.
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• Example A card is dealt. What is the
  probability it is either a queen or a spade?

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• Example True or false? In a shuffled deck of
  cards
• A) The chance that the top card is the jack of
  clubs is 1/52.
•  
• B) The chance that the bottom card is eight of
  hearts is 1/52.
• C) The chance that the top card is the jack
  of clubs or the bottom card is the eight of
  hearts is 2/52.

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• D) The chance that the top card is the jack of
  clubs or the bottom card is the jack of clubs is
  2/52.
•   
• E) The chance that the top card is the jack of
  clubs and the bottom card is the eight of hearts
  is
• F) The chance that the top card is the jack
  of clubs and the bottom card is the jack of clubs
  is

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• Example Four tickets are drawn from a box that
  contains 5 tickets each numbered from 1 to 5.
  Find P(at least one 2) if the draws are made

A) with replacement B) without replacement
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• Example Suppose that P(A) 0.2, P(B) 0.3
  and 0.4. Use a Venn
  diagram to find the following quantities
• 1.
• 2.
• 3.
• 4.

P(A or B)
P(A n B )
P(A n B )
P(A n B )
P(A B)
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Multiplication Trees and Bayes Rule
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• Example Suppose there are two electrical
  components. The chance that the first component
  fails is 20. If the first fails, the chance that
  the second fails is 30. However, if the first
  works, the chance that the second fails is 10.
• Find the probability that
• 1. At least one component works.
• 2. Exactly one component works.
• 3. The second component works.

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• Solution A multi-stage problem like this can be
  visualized with a tree diagram

Works
0.72
0.9
Works
0.8
0.1
Fails
0.08
Works
0.14
0.2
0.7
Fails
0.3
Fails
0.06
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Works
0.72
0.9
Works
0.8
0.1
Fails
0.08
Works
0.14
0.2
0.7
Fails
0.3
Fails
0.06
P(at least one component works) P(exactly one
component works) P(second component works)
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• Example In a class of 50 students, what is the
  probability that at least two share a birthday?

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• Example In Barcelona, the local men tend not to
  be overweight due to genetic and dietary reasons.
  However, the tourists from around the world who
• visit Barcelona tend to be as overweight as the
  rest of Europe and America.
• Suppose
• 2 of Barcelona men are overweight
• 35 of Barcelonas male tourists are overweight
  
• 10 of the men in Barcelona are tourists
• If a man is selected at random, what is the
  probability that hes overweight?

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• Example Suppose 2 of Barcelona men are
  overweight, while 35 of Barcelonas male
  tourists are overweight. Also, suppose 10 of the
  men in Barcelona are tourists.
• If you see an overweight man at a distance in
  Barcelona, what is the probability that hes a
  tourist?
• Solution Let A man is overweight, B man
  is a tourist.
• Then we are given
• P(A B) 0.35
• P(A B ) 0.02
• P(B) 0.1
• But we want P(B A) the reverse conditional
  probability.

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• Example In a study of 101 patients, 37 do not
  have coronary artery disease (CAD) and 64 have
  CAD. All 101 patients were given a certain test
  for CAD. Of the 37 patients without CAD, 34 had a
  negative test while 3 had a positive test. Of the
  64 patients with CAD, 54 had a positive test and
  10 had a negative test.
• A man in his 50s with a family history of CAD
  sees a doctor, complaining of chest pain. Because
  of his age and family history, the doctor
  estimates the probability that the patient has
  CAD as 0.6. The patient is then given the above
  test, and it comes back positive. Find the
  posterior probability that the patient has CAD.

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• Solution We begin with the tree diagram

Test positive
0.506
54/64
Disease positive
Test negative
0.6
0.094
10/64
Test positive
0.032
3/37
0.4
Disease negative
Test negative
0.368
34/37
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• Definition
• Sensitivity P(test positive disease
  positive)
• Specificity P(test negative disease
  negative)
• In a perfect world, both of these conditional
  probabilities would be 100, but thats
  unrealistic. Bayes rule allows us to take the
  information from a clinical trial and use it in a
  diagnostic setting.
• Measuring these two quantities is absolutely
  essential in a clinical medical trial. Lets look
  up how many articles have been published which
  contain these words in the abstract.

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• Example The receptor interleukin-8 is measured
  as a test for ectopic (tubal) pregnancy. For 17
  women with ectopic pregnancy, 14 tested positive
  and 3 tested negative. For 55 women without an
  ectopic pregnancy, 10 tested positive and 45
  tested negative.
• (a) Find the sensitivity and specificity of the
  test.
• (b) An obstetrician treats a woman with mild
  pelvic pain. Based on her professional
  experience, the doctor believes the probability
  of an ectopic pregnancy to be 0.2. The patient is
  tested, and the test returns negative. Find the
  posterior probability of an ectopic pregnancy.

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