Probability: The Mathematics of Chance

1
Chapter 7

• Probability The Mathematics of Chance

2
Probability

• Random if the individual outcomes are uncertain,
  but the long term behavior is predictable, then
  it is called random

3
Probability Theory

• Describes the long term behavior of an action
• Probability the proportion of an outcome in a
  very long run of trials of an experiment

4
Flipping A Coin
5
Terminology

• Experiment an activity or phenomenon that is
  under consideration
• Outcome the results of an experiment

6
Terminology

• Trial one observation of the experiment
• Sample space the set of all possible outcomes
  from an experiment

7
More Terminology

• Sample point or simple outcome each element of
  the sample space
• Event A subset of a sample space

8
Rolling Dice
9
Probability Rules

• 1 every probability, p(s), is a number between
  0 and 1
• 2 the sum of the probabilities, p(s), for all
  outcomes, s, in a sample space, S, is exactly 1

10
Probability Rules

• 3 if two events have no outcomes in common,
  then the probability that one or the other occurs
  is the sum of their probabilities

11
Example

• Roll two diceWhat is the probability of rolling
  a five?, An even number?, A number greater than
  2?, A prime number?

12
Probability Histogram
13
Probability Model

• A probability model is a mathematical description
  of an event
• It contains a sample space and a manner to assign
  probabilities to every event

14
Household Size

• Verify the probability model for the following
  data

15
Equally Likely Outcomes

• If a random experiment has k possible outcomes,
  all equally likely, then each outcome has a
  probability 1/k

16
Equally Likely Outcomes
17
Compound Events

• When in an equally likely experiment, the
  probability of an event A is

18
Example

• If you were given a secret code consisting of
  three letters, what is the probability of getting
  one with no x's?

19
Example

• If the letters cannot repeat, what is the
  probability of getting no xs?

20
Counting Rule A

• Suppose you have n things, if you want to arrange
  k of then with repetition, then there would be n
  ? n ? ? ? n nk

21
Counting Rule B
Suppose you have n things, if you want to arrange
k of then without repetition, then there would
be
22
Example

• Seven students are selected to judge a speaking
  competition. In how many ways can they be seated
  on a stage with seven chairs?

23
Two Games

• Game 1
• 1/10 chance of winning 10,000
• 9/10 chance of winning nothing

• Game 2
• ½ chance of winning 10
• ½ chance of winning nothing

24
Mean of Random Phenomenon

• The mean of a probability model is given by the
  following model

25
Household Size

• Find the mean for this probability model

26
Household Size
27
Sum of Two Dice
28
Law of Large Numbers

• Observe any random phenomenon having numerical
  outcomes with finite mean ?.Two things will
  happen
• The proportion of trials on which any outcome
  occurs will approach the probability for that
  outcome, will approach ?

29
Sampling Distributions

• Sampling variability many samples from the same
  population will yield different results
• Using histogram to show probability

30
Small Sample Size
31
Large Sample Size
32
Statistics

• A number produced from a sample is a statistic
• Sampling distribution the distribution of values
  taken by the statistic in all possible samples
  from a population

33
Normal Distributions

• A normal curve assigns probabilities to outcomes
  by the measure of area under the curve associated
  with a given outcome

34
Normal Distributions
35
1493 Sample Size
36
Shape of Normal Curves

• The mean of a normal distribution lies at the
  center of the symmetry of the curve
• The standard deviation is located approximately
  where the curve changes from down to out

37
Normal Curve
38
Quartiles of a Normal Distribution

• Q1 is located 0.67? below the mean
• Q3 is located 0.67? above the mean
• Remember the median is the mean in a normal
  distribution

39
Womens Heights
40
The 68-95-99.7 Rule

• 68 of all observations fall within one standard
  deviation of the mean
• 95 of all observations fall within two standard
  deviation of the mean
• 99.7 of all observations fall within three
  standard deviation of the mean

41
The 68-95-99.7 Rule
42
Outside the 2nd Deviation
43
The Central Limit Theorem

• A sample mean, or sample proportion from n trials
  of an experiment has a distribution that is
  approximately normal when n is large

44
The Central Limit Theorem

• The mean of the normal distribution is the same
  as a mean of a single trial

45
The Central Limit Theorem

• The standard deviation of the normal distribution
  is the standard deviation of a single trial
  divided by

46
Casino Example
47
What About Finding the Standard Deviation?

• The standard deviation for a probability model
  can be found from the variance