Understanding

1
Chapter 15

• Understanding
• Probability
• and
• Long-Term Expectations

2
Thought Questions
1, page 256

• Here are two very different queries about
  probability
• If you flip a coin and do it fairly, what is the
  probability that it will land with heads up?
• What is the probability that you will eventually
  own a home, i.e. how likely do you think it is?
  (If you already own a home, what is the
  probability that you will own a different home in
  the next five years?)
• For which question was it easier to provide a
  precise answer? Why?

3
Thought Questions
Supplemental (from Seeing Through Statistics,
First Edition)

• Which of the following more closely describes
  what it means to say that the probability of a
  tossed coin landing with heads up is ½?
• After more and more tosses, the fraction of heads
  will get closer and closer to ½.
• The number of heads will always be about half of
  the number of tosses.

4
Thought Questions
3, page 256
Explain what is wrong with the following
statement, given by a student as a partial answer
to Thought Question 1 The probability that I
will eventually own a home, or of any other
particular event happening, is 1/2 because either
it will happen or it wont.
5
Thought Questions
5, page 256
How much would you be willing to pay for a ticket
to a contest in which there was a 1 chance that
you would win 500 and a 99 chance you would win
nothing? Explain your answer.
6
Two Concepts of Probability

• Personal-Probability Interpretation
• The degree to which a given individual believes
  the event in question will happen.
• Personal belief
• Relative-Frequency Interpretation
• The proportion of time the event in question
  occurs over the long run.
• Long-run relative frequency

7
Relative-Frequency Probabilities

• Two ways to determine
• Physical assumptions (theoretical mathematical
  model)
• Repeated observations (empirical results)
• Experience with many samples
• Simulation

8
Relative-Frequency Probabilities Summary

• Can be applied when the situation can be repeated
  numerous times (conceptually) and the outcome can
  be observed each time.
• Relative frequency of an outcome settles down to
  one value over the long run. That one value is
  then defined to be the probability of that
  outcome.
• The probability cannot be used to determine
  whether or not the outcome will occur on a single
  occasion.

9
Personal or Relative Frequency Probabilities?

• The probability that a lottery ticket will be a
  winner.
• The probability that you will get a B in this
  course.
• The probability that a randomly selected student
  in one of your professors classes will get a B.

10
Personal or Relative Frequency Probabilities?

• The probability that the 7 a.m. flight from San
  Francisco to New York will be on time on a
  randomly selected day.
• The probability that the Atlanta Braves
  professional baseball team will win the World
  Series in the year 2001.

11
Probability Rules

• Four simple, logical rules which apply to how
  probabilities relate to each other and to real
  events.
• Review the rules in section 15.4, pages 261-264.
• Review the ten examples given on these pages!!

12
Probability Rule 1

• If there are only two possible outcomes in an
  uncertain situation, then their probabilities
  must add to one.
• As a jury member, you assess the probability that
  the defendant is guilty to be 0.80. Thus you
  must also believe the probability the defendant
  is not guilty is 0.20 in order to be coherent
  (consistent with yourself).
• If the probability that a flight will be on time
  is .70, then the probability it will be late is
  .30.

13
Probability Rule 2

• If two outcomes cannot happen simultaneously,
  they are said to be mutually exclusive. The
  probability of one or the other of two mutually
  exclusive outcomes happening is the sum of their
  individual probabilities.
• Age of woman at first child birth
• under 20 25
• 20-24 33
• 25 ?

24 or younger 58
Rule 1 42
14
Probability Rule 3

• If two events do not influence each other, and
  if knowledge about one doesnt help with the
  knowledge of the probability of the other, the
  events are said to be independent of each other.
  If two events are independent, the probability
  that they both happen is found by multiplying
  their individual probabilities.

15
Probability Rule 3 Example

• Suppose that about 20 of incoming male freshmen
  smoke.
• Suppose that these freshmen are randomly assigned
  in pairs to dorm rooms.
• The probability of a match (both smokers or both
  non-smokers)
• both are smokers 0.04 (0.20)(0.20)
• neither is a smoker 0.64 (0.80)(0.80)
• only one is a smoker ?

68
Rule 1 32
What if pairs are self-selected?
16
Probability Rule 4

• If the ways in which one event can occur are a
  subset of those in which another event can occur,
  then the probability of the first event cannot be
  higher that the probability of the one for which
  it is a subset.
• Suppose you see an elderly couple and you think
  the probability that they are married is 80.
• Suppose you think the probability that the
  elderly couple is married with children is 95.
• These two personal probabilities are not
  coherent. Why?

17
Probability Rule 4
Probability of married with children must not be
greater than the probability that the couple is
married.
18
The Main Point...
Long-Term Gains, Losses and Expectations

• While we cannot predict individual outcomes, we
  can predict what happens (on average) in the long
  run.

19
Long-Term Gains, Losses and Expectations

• Tickets to a school fund-raiser event sell for
  1.
• One ticket will be randomly chosen, the ticket
  owner receives 500.
• They expect to sell 1,000 tickets. Your ticket
  has a 1/1000 probability of winning.
• Two outcomes
• You win 500, net gain is 499.
• You do not win, net gain is -1.

20
Expected Value

• Your expected gain (expected value) is
  (499)(0.001) (-1)(0.999) -0.50.
• long term, you loose an average of 0.50 each
  time (conceptually) you enter such a contest.
• Hey, the school needs to make a profit!

21
Make a Decision, Which Do You Choose?
(1) A gift of 240, guaranteed. (2) A 25 chance
to win 1,000 and a 75 of getting nothing.

• First alternative EV240, no variation.
• Second alternative EV(1000)(0.25)
  (0)(0.75) 250
• Make a Decision

22
Make a Decision, Which Do You Choose?
(1) A gift of 240, guaranteed. EV240 (2) A
25 chance to win 1,000 and a 75 of getting
nothing. EV250

• If choosing for ONE trial
• option (2) will maximize potential gain (1000)
• option (2) will maximize expected gain
• option (1) guarantees a gain
• If choosing for MANY (500?) trials
• option (2) will maximize expected gain(will make
  more money in the long run)

23
Make a Decision, Which Do You Choose?
(1) A sure loss of 740. (2) A 75 chance to lose
1,000 and a 25 to lose nothing.

• First alternative EV740, no variation.
• Second alternative EV(1000)(0.75)
  (0)(0.25) 750
• Make a Decision

24
Make a Decision, Which Do You Choose?
(1) A sure loss of 740. EV740 (2) A 75
chance to lose 1,000 and a 25 to lose nothing.
EV750

• If choosing for ONE trial
• option (2) will minimize potential loss (0)
• option (1) will minimize expected loss
• option (1) guarantees a loss
• If choosing for MANY (500?) trials
• option (1) will minimize expected loss (will
  lose less money in the long run)

25
Key Concepts

• Personal probability.
• Long-run Relative Frequency interpretation of
  probability.
• Rules for probability.
• Probability can be used to make accurate
  predictions about long-run averages and events.