Introducing Probability

1
Chapter 9

• Introducing Probability

2
Idea of probability

• Probability is the science of chance behavior
• Chance behavior is unpredictable in the short run
  but has a predictable pattern in the long run

3
Randomness and probability

• A phenomenon is random if individual outcomes are
  uncertain, but there is nonetheless a regular
  distribution of outcomes in a large number of
  repetitions.
• The probability of any outcome of a random
  phenomenon can be defined as the proportion of
  times the outcome would occur in a very long
  series of repetitions.

4
Thinking about probabilities

• The best way to understand randomness is to
  observe random behavior in a long run of
  independent trials
• Short runs give only rough estimates of
  probability

5
Empirical probabilities
Coin flipping eventually, the proportion
approaches 0.5, the probability of a head
6
Exercise 9.5 (p. 227)

• Premise Probability of 0 in the random number
  table is 0.1
• What proportion of the first 50 digits is a 0?
  (ans 3 of 50, or 0.06)
• Use the Probability Applet to simulate 40 at a
  time set probability to 0.1. What is the result
  of 200 tosses?
• C\Data\hs067\BPS3e\index.htm

7
Probability Models

• Skip this section
• (pp. 228 230)

8
Probability Rule 1

• Any probability is a number between 0 and 1.
• A probability can be interpreted as the
  proportion of times that a certain event can be
  expected to occur.
• If the probability of an event is more than 1,
  then it will occur more than 100 of the time
  (Impossible!).

9
Probability Rule 2

• All possible outcomes together must have
  probability 1.
• Because some outcome must occur on every trial,
  the sum of the probabilities for all possible
  outcomes must be exactly one.
• If the sum of all of the probabilities is less
  than one or greater than one, then the resulting
  probability model will be incoherent.

10
Probability Rule 3

• The probability that an event does not occur is
  1 minus the probability that the event does
  occur.
• If a person has a 0.75 chance of recovering, she
  must have a 1 0.75 0.250 chance of not
  recovering.
• If a person has a 0.95 chance of recovering, she
  must have a 1 0.95 0.05 chance of not
  recovering.

11
Probability Rule 4

• If two events have no outcomes in common, they
  are said to be disjoint. The probability that
  one or the other of two disjoint events occurs 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 3 (or 2) 42
12
Probability RulesMathematical Notation
13
Assigning probabilities finite

• Skip this section (pp. 232 235)

14
Assigning probabilities intervals

• Recall areas under a density curve (Chapter
  3)!
• Illustration random number generators give
  output (digits) spread uniformly across the
  interval from 0 to 1.

Find the probability of getting a random number
that is less than or equal to 0.5 OR greater than
0.8.
P(X 0.5 or X gt 0.8) P(X 0.5) P(X gt 0.8)
0.5 0.2 0.7
15
Normal probability models

• The Normal curve ? the density curve that is most
  familiar to us
• Normal random variable denoted X N(µ, ?)
• Technique for finding Normal probabilities
  covered in Chapter 3
• Convert observed values of the endpoints of the
  interval to Z scores
• Find probabilities from Table A
• Example 9.9 in text (p. 237)

16
Random variable and Personal Probabilities

• Skip these sections (pp. 237 241)