Warm-up

1
Warm-up

• Define the sample space of each of the following
  situations
• Choose a student in your class at random. Ask
  how much time that student spent studying during
  the past 24 hours.
• The Physicians Health Study asked 11,000
  physicians to take an aspirin every other day and
  observed how many of them had a heart attack in a
  five-year period.
• In a test of new package design, you drop a
  carton of a dozen eggs from a height of 1 foot
  and count the number of broken eggs.

2
Section 5.2

• Independence and the Multiplication Rule

3
Two Special Rules

• Weve learned the addition rule for disjoint
  events If A and B are disjoint, then P(A or B)
  P(A) P(B).
• Now well learn the multiplication rule for
  independent events that if A and B are
  independent, then P(A and B) P(A)?P(B)
• Remember that two events are independent if
  knowing that one occurs does not change the
  probability that the other occurs.

4
Examples of Independent Events

• Toss a coin twice. Let A first toss is a head
  and B second toss is a tail. Events A and B
  are independent. Thus, the P(AnB) P(A)P(B).
• P(Head and Tail) P(Head) P(Tail) ½ ½ ¼
• Draw 3 cards from a deck, replacing and shuffling
  in between each draw. This is called with
  replacement.

5
Without Replacement

• If you draw three cards from a deck without
  replacing, the probabilities change on each draw.
  Therefore, drawing without replacement is NOT
  independent.

6
Cautions

• You must be told or have prior knowledge that an
  event is disjoint or independent.
• Do not confuse disjoint with independent.
  Disjoint can be displayed in a Venn Diagram.
  Independence can not.

• The addition rule for disjoint events and the
  multiplication rule for independent events only
  work when the criteria are met. Resist the
  temptation to use them for events that are not
  disjoint or not independent.

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Sample Questions

• Suppose that among the 6000 students at a high
  school, 1500 are taking honors courses and 1800
  prefer watching basketball to watching football.
  If taking honors courses and preferring
  basketball are independent, how many students are
  both taking honors courses and prefer basketball
  to football?

8
Sample Questions

• Suppose that for any given year, the
  probabilities that the stock market declines is
  .4, and the probability that womens hemlines are
  lower is .35. Suppose that the probability that
  both events occur is .3. Are the two events
  independent?

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Sample Questions

• In a 1974 Dear Abby letter, a woman lamented
  that she had just given birth to her eighth
  child, and all were girls! Her doctor had
  assured her that the chance of the eighth child
  being a girl was only 1 in 100.
• A) What was the real probability that the eighth
  child would be a girl?
• B) Before the birth of the first child, what was
  the probability that the woman would give birth
  to eight girls in a row?

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Section 6.3General Probability Rules
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Special Probability Rules

• If A and B are disjoint, then P(AUB) P(A)
  P(B).
• This rule relies on a condition to be met.
• So, what if the events are NOT disjoint?
• What if there are more than 2 disjoint events?

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General Probability Rules

• P(AUB) P(A) P(B) P(AnB)
• This is how the formula will appear on your
  formula sheet for the exam (and thus my tests).

If A and B are disjoint, then what does P(AnB)
equal?
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Sample Question Student Survey

• (making an A in Statistics only) 18 students
  (making an A in
  Calculus only) 63 students
  (making an A in Stats and in Calculus)
  27 students
• S150 students
• Draw a Venn diagram.
• Find the probability of making an A in Stats but
  not in Calc.
• Find the probability of making an A in Calc but
  not in Stats.
• Find the probability of not making an A in either
  subject.
• Find the probability of making an A in either
  Calc or Stats.
• Are these events independent?

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Extending the Rules to Three Events

• If you have three disjoint events, then P(A or B
  or C) P(A) P(B) P(C)
• If they are not disjoint, then P(A or B or C)
  P(A) P(B) P(C) P(AnB) P(AnC) P(BnC)
• Often it is easier to draw a Venn Diagram than to
  use the formula.
• Look at p365 6.51

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Sample Questions

• Suppose that the probability that you will
  receive an A in AP Statistics is .35, the
  probability that you will receive As in both AP
  Statistics and AP Biology is .19, and the
  probability that you will receive an A in AP Bio
  but not in Stats is .17. Which is a proper
  conclusion?
• A) P(A in AP Bio) .36
• B) P(you didnt take Bio) .01
• C) P(not making an A in AP Stat or Bio) .38
• D) The given probabilities are impossible.

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Sample Questions

• If P(A) .2 and P(B) .1, what is P(AUB) if A
  and B are independent?
• A) .02
• B) .28
• C) .30
• D) .32
• E) There is insufficient information to answer
  this question.

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Sample Questions

• Given the probabilities P(A) .4 and P(AUB)
  .6, what is the probability P(B) if A and B are
  mutually exclusive? If A and B are independent?
• A) .2, .28
• B) .2, .33
• C) .33, .2
• D) .6, .33
• E) .28, .2

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Conditional Probability

• When events are not independent, the probability
  of one changes if we know that the other event
  occurred.
• I will draw two cards without replacement.
• Let A 1st card I draw is an Ace
• Let B 2nd card I draw is an Ace
• The probability of B occurring changes depending
  on whether A occurred.
• The new notation P(BA) is read the probability
  of B given A. It asks you to find the
  probability of B knowing that A has occurred.

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Lets look at education and age
Education Age Age Age Age
Education 25 34 35 54 55 Total
No high school 4,474 9,155 14,224 27,853
Completed HS 11,546 26,481 20,060 58,087
1 to 3 years college 10,700 22,618 11,127 44,445
4 years of college 11,066 23,183 10,596 44,845
Total 37,786 81,435 56,008 175,230
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Find these probabilities

• Let A the person chosen is 25 34
• Let B the person chosen has 4 years of
  college
• Find P(A).
• Find P(A and B).
• Find P(BA).
• Given that a person has only a HS diploma, what
  is the probability that the person is 55 or
  older?

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The general rule for multiplication

• P(AnB) P(A)P(BA)
• This can be rewritten as
• P(BA) P(AnB)/P(A)
• Example Two cards are drawn without
  replacement.
• Let A 1st card is a spade
• Let B 2nd card is a spade
• Find P(BA).
• Find P(A and B).

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Next Example

• Suppose the probability that the dollar falls in
  value compared to the yen is 0.5.
• Suppose that the probability that if the dollar
  falls in value, a supplier from Japan will
  renegotiate their contract is 0.7.
• What is the probability that the dollar will fall
  in value and the supplier demands renegotiation?

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Homework

• Chapter 5 4, 9, 40, 43, 51, 60, 83, 89, 94