1. Probability and Probability Distributions
Part III
Probability and
Probability Distributions

2. Chapter 5
Probability

3. Chapter 5
When we talk about probability, we are talking about a (mathematical) measure of how likely it is for some particular thing to happen
Probability deals with chance behavior
We study outcomes, or results of experiments
Each time we conduct an experiment, we may get a different result
Probability models the short-term behavior of experiments

4. Chapter 5 – Section 1
Rule – the probability of any event must be greater than or equal to 0 and less than or equal to 1
It does not make sense to say that there is a –30% chance of rain
It does not make sense to say that there is a 140% chance of rain
Note – probabilities can be written as decimals (0, 0.3, 1.0), or as percents (0%, 30%, 100%), or as fractions (3/10)

5. Chapter 5 – Section 1
If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by
This becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers)

6. Chapter 5 – Section 1
Sometimes probabilities are difficult to calculate, but the experiment can be simulated on a computer
If we simulate the experiment multiple times, then this is similar to the situation for the empirical method
We can use

7. Chapter 5 – Section 1 Example
We wish to determine what proportion of students at a certain school have type A blood
We perform an experiment (a simple random sample!) with 100 students
If 29 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 29%

8. Chapter 5
Descriptive statistics, describing and summarizing data, deals with data as it is
Probability , modeling data, deals with data as it is predicted to be
The combination of the two will let us do our inferential statistics techniques in Part IV

9. Discrete Probability Distributions
Chapter 6
Discrete Probability
Distributions

10. Overview
These are probability distributions that are designed to model discrete variables
Many of the discrete probability distributions model “counts”

11. Chapter 6 Sections Sections in Chapter 6 Discrete Random Variables
The Binomial Probability Distribution
6.1
6.2

12. Discrete Random Variables
Chapter 6 Section 1
Discrete
Random Variables

13. Chapter 6 – Section 1 Learning objectives
Distinguish between discrete and continuous random variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random variable
Interpret the mean of a discrete random variable as an expected value
Compute the variance and standard deviation of a discrete random variable 1 2 3 4 5 6

14. Chapter 6 – Section 1 Learning objectives
Distinguish between discrete and continuous random variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random variable
Interpret the mean of a discrete random variable as an expected value
Compute the variance and standard deviation of a discrete random variable 1 2 3 4 5 6

15. Chapter 6 – Section 1
A random variable is a numeric measure of the outcome of a probability experiment
Random variables reflect measurements that can change as the experiment is repeated
Random variables are denoted with capital letters, typically using X (and Y and Z …)
Values are usually written with lower case letters, typically using x (and y and z ...)

16. Chapter 6 – Section 1 Examples
Tossing four coins and counting the number of heads
The number could be 0, 1, 2, 3, or 4
The number could change when we toss another four coins
Measuring the heights of students
The heights could change from student to student
Examples
Tossing four coins and counting the number of heads
The number could be 0, 1, 2, 3, or 4
The number could change when we toss another four coins

17. Chapter 6 – Section 1
A discrete random variable is a random variable that has either a finite or a countable number of values
A finite number of values such as {0, 1, 2, 3, and 4}
A countable number of values such as {1, 2, 3, …}
Discrete random variables are designed to model discrete variables (see section 1.2)
Discrete random variables are often “counts of …”
A discrete random variable is a random variable that has either a finite or a countable number of values
A finite number of values such as {0, 1, 2, 3, and 4}
A countable number of values such as {1, 2, 3, …}

18. Chapter 6 – Section 1 An example of a discrete random variable
The number of heads in tossing 3 coins (a finite number of possible values)
There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads
A finite number of possible values – a discrete random variable
This fits our general concept that discrete random variables are often “counts of …”
An example of a discrete random variable
The number of heads in tossing 3 coins (a finite number of possible values)
There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads
A finite number of possible values – a discrete random variable
An example of a discrete random variable
The number of heads in tossing 3 coins (a finite number of possible values)
There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads
An example of a discrete random variable
The number of heads in tossing 3 coins (a finite number of possible values)

19. Chapter 6 – Section 1 Other examples of discrete random variables
The possible rolls when rolling a pair of dice
A finite number of possible pairs, ranging from (1,1) to (6,6)
The number of pages in statistics textbooks
A countable number of possible values
The number of visitors to the White House in a day
Other examples of discrete random variables
The possible rolls when rolling a pair of dice
A finite number of possible pairs, ranging from (1,1) to (6,6)
The number of pages in statistics textbooks
A countable number of possible values
Other examples of discrete random variables
The possible rolls when rolling a pair of dice
A finite number of possible pairs, ranging from (1,1) to (6,6)
Other examples of discrete random variables

20. Chapter 6 – Section 1
A continuous random variable is a random variable that has an infinite, and more than countable, number of values
The values are any number in an interval
Continuous random variables are designed to model continuous variables (see section 1.1)
Continuous random variables are often “measurements of …”
A continuous random variable is a random variable that has an infinite, and more than countable, number of values
The values are any number in an interval

21. Chapter 6 – Section 1 An example of a continuous random variable
The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit
The possible values (assuming that we can measure temperature to great accuracy) are in an interval
The interval may be something like (–20,110)
This fits our general concept that continuous random variables are often “measurements of …”
An example of a continuous random variable
The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit
The possible values (assuming that we can measure temperature to great accuracy) are in an interval
The interval may be something like (–20,110)
An example of a continuous random variable
The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit
The possible values (assuming that we can measure temperature to great accuracy) are in an interval
An example of a continuous random variable
The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit

22. Chapter 6 – Section 1 Other examples of continuous random variables
The height of a college student
A value in an interval between 3 and 8 feet
The length of a country and western song
A value in an interval between 1 and 15 minutes
The number of bytes of storage used on a 80 GB (80 billion bytes) hard drive
Although this is discrete, it is more reasonable to model it as a continuous random variable between 0 and 80 GB
Other examples of continuous random variables
The height of a college student
A value in an interval between 3 and 8 feet
The length of a country and western song
A value in an interval between 1 and 15 minutes
Other examples of continuous random variables
The height of a college student
A value in an interval between 3 and 8 feet
Other examples of continuous random variables

23. Chapter 6 – Section 1
The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities
A distribution could be
In the form of a table
In the form of a graph
In the form of a mathematical formula
The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities

24. Chapter 6 – Section 1
If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x
Examples
In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5
In rolling one die, if X is the number rolled, then P(1) = 1/6
If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x

25. Chapter 6 – Section 1 Properties of P(x)
Since P(x) form a probability distribution, they must satisfy the rules of probability
0 ≤ P(x) ≤ 1
Σ P(x) = 1
In the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

26. Chapter 6 – Section 1
An example of a discrete probability distribution
All of the P(x) values are positive and they add up to 1 x
P(x) 1 .2 2 .6 5 .1 6

27. Chapter 6 – Section 1
An example that is not a probability distribution
Two things are wrong
P(5) is negative
The P(x)’s do not add up to 1
An example that is not a probability distribution
Two things are wrong
P(5) is negative
An example that is not a probability distribution
Two things are wrong x
P(x) 1 .2 2 .6 5
-.3 6 .1 x
P(x) 1 .2 2 .6 5
-.3 6 .1 x
P(x) 1 .2 2 .6 5
-.3 6 .1

28. Chapter 6 – Section 1 A probability histogram is a histogram where
The horizontal axis corresponds to the possible values of X (i.e. the x’s)
The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s)
A probability histogram is very similar to a relative frequency histogram

29. Chapter 6 – Section 1 An example of a probability histogram
The histogram is drawn so that the height of the bar is the probability of that value

30. Chapter 6 – Section 1
The mean of a probability distribution can be thought of in this way:
There are various possible values of a discrete random variable
The values that have the higher probabilities are the ones that occur more often
The values that occur more often should have a larger role in calculating the mean
The mean is the weighted average of the values, weighted by the probabilities
The mean of a probability distribution can be thought of in this way:
There are various possible values of a discrete random variable
The values that have the higher probabilities are the ones that occur more often
The values that occur more often should have a larger role in calculating the mean
The mean of a probability distribution can be thought of in this way:
There are various possible values of a discrete random variable
The values that have the higher probabilities are the ones that occur more often
The mean of a probability distribution can be thought of in this way:
There are various possible values of a discrete random variable

31. Chapter 6 – Section 1 The mean of a discrete random variable is
μX = Σ [ x • P(x) ]
In this formula
x are the possible values of X
P(x) is the probability that x occurs
Σ means to add up these terms for all the possible values x
The mean of a discrete random variable is
μX = Σ [ x • P(x) ]

32. Chapter 6 – Section 1 Example of a calculation for the mean
Add: = 2.5
The mean of this discrete random variable is 2.5
Example of a calculation for the mean
Example of a calculation for the mean x
P(x)
x • P(x) 1
0.2 2
0.6
1.2 5
0.1
0.5 6 x
P(x) 1
0.2 2
0.6 5
0.1 6 x
P(x)
x • P(x) 1
0.2 2
0.6 5
0.1 6
Multiply
Multiply
Multiply again
Multiply again

33. Chapter 6 – Section 1 The calculation for this problem written out
μX = Σ [ x • P(x) ]
= [1• 0.2] + [2• 0.6] + [5• 0.1] + [6• 0.1] =
= 2.5
The mean of this discrete random variable is 2.5

34. Chapter 6 – Section 1
The mean can also be thought of this way (as in the Law of Large Numbers)
If we repeat the experiment many times
If we record the result each time
If we calculate the mean of the results (this is just a mean of a group of numbers)
Then this mean of the results gets closer and closer to the mean of the random variable
The mean can also be thought of this way (as in the Law of Large Numbers)
If we repeat the experiment many times
If we record the result each time
If we calculate the mean of the results (this is just a mean of a group of numbers)
The mean can also be thought of this way (as in the Law of Large Numbers)
If we repeat the experiment many times
The mean can also be thought of this way (as in the Law of Large Numbers)
The mean can also be thought of this way (as in the Law of Large Numbers)
If we repeat the experiment many times
If we record the result each time

35. Chapter 6 – Section 1
The expected value of a random variable is another term for its mean
The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable
The expected value of a random variable is another term for its mean

36. Chapter 6 – Section 1
The variance of a discrete random variable is computed similarly as for the mean
The mean is the weighted sum of the values
μX = Σ [ x • P(x) ]
The variance is the weighted sum of the squared differences from the mean
σX2 = Σ [ (x – μX)2 • P(x) ]
The standard deviation, as we’ve seen before, is the square root of the variance … σX = √ σX2
The variance of a discrete random variable is computed similarly as for the mean
The mean is the weighted sum of the values
μX = Σ [ x • P(x) ]
The variance is the weighted sum of the squared differences from the mean
σX2 = Σ [ (x – μX)2 • P(x) ]
The variance of a discrete random variable is computed similarly as for the mean
The mean is the weighted sum of the values
μX = Σ [ x • P(x) ]
The variance of a discrete random variable is computed similarly as for the mean

37. Chapter 6 – Section 1 The variance formula
σX2 = Σ [ (x – μX)2 • P(x) ]
can involve calculations with many decimals or fractions
An equivalent formula is
σX2 = [ Σ x2 • P(x) ] – μX2
This formula is often easier to compute

38. Chapter 6 – Section 1
For variables and samples (section 3.2), we had the concept of a population variance (for the entire population) and a sample variance (for a sample from that population)
These probability distributions model the complete population
These are population variance formulas
There is no analogy for sample variance here

39. Chapter 6 – Section 1
The variance can be calculated by hand, but the calculation is very tedious
Whenever possible, use technology (calculators, software programs, etc.) to calculate variances and standard deviations

40. Summary: Chapter 6 – Section 1
Discrete random variables are measures of outcomes that have discrete values
Discrete random variables are specified by their Discrete probability distributions
The mean of a discrete random variable can be interpreted as the long term average of repeated independent experiments
The variance of a discrete random variable measures its dispersion from its mean

41. The Binomial Probability Distribution
Chapter 6 Section 2
The Binomial
Probability Distribution

42. Chapter 6 – Section 2
A binomial experiment has the following structure
The first test is performed … the result is either a success or a failure
The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same
A third test is performed … the result is either a success or a failure. The result is independent of the first two and the chance of success is the same
A binomial experiment has the following structure
The first test is performed … the result is either a success or a failure
The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same
A binomial experiment has the following structure
The first test is performed … the result is either a success or a failure
A binomial experiment has the following structure

43. Chapter 6 – Section 2 Example Example Example Example Example
A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit
The card is then put back into the deck
A second card is drawn from the deck with the same definition of success.
The second card is put back into the deck
We continue for 10 cards
Example
A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit
The card is then put back into the deck
A second card is drawn from the deck with the same definition of success.
The second card is put back into the deck
Example
A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit
The card is then put back into the deck
Example
A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit
Example
A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit
The card is then put back into the deck
A second card is drawn from the deck with the same definition of success.

44. Chapter 6 – Section 2
A binomial experiment is an experiment with the following characteristics
The experiment is performed a fixed number of times, each time called a trial
The trials are independent
Each trial has two possible outcomes, usually called a success and a failure
The probability of success is the same for every trial
A binomial experiment is an experiment with the following characteristics
The experiment is performed a fixed number of times, each time called a trial
The trials are independent
Each trial has two possible outcomes, usually called a success and a failure
A binomial experiment is an experiment with the following characteristics
The experiment is performed a fixed number of times, each time called a trial
A binomial experiment is an experiment with the following characteristics
A binomial experiment is an experiment with the following characteristics
The experiment is performed a fixed number of times, each time called a trial
The trials are independent

45. Chapter 6 – Section 2 Notation used for binomial distributions
The number of trials is represented by n
The probability of a success is represented by p
The total number of successes in n trials is represented by X
Because there cannot be a negative number of successes, and because there cannot be more than n successes (out of n attempts)
0 ≤ X ≤ n

46. Chapter 6 – Section 2 In our card drawing example
Each trial is the experiment of drawing one card
The experiment is performed 10 times, so n = 10
The trials are independent because the drawn card is put back into the deck
Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else
The probability of success is 0.25, the same for every trial, so p = 0.25
X, the number of successes, is between 0 and 10
In our card drawing example
Each trial is the experiment of drawing one card
The experiment is performed 10 times, so n = 10
The trials are independent because the drawn card is put back into the deck
Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else
The probability of success is 0.25, the same for every trial, so p = 0.25
In our card drawing example
Each trial is the experiment of drawing one card
The experiment is performed 10 times, so n = 10
The trials are independent because the drawn card is put back into the deck
In our card drawing example
Each trial is the experiment of drawing one card
The experiment is performed 10 times, so n = 10

47. Chapter 6 – Section 2
We would like to calculate the probabilities of X, i.e. P(0), P(1), P(2), …, P(n)
Do a simpler example first
For n = 3 trials
With p = .4 probability of success
Calculate P(2), the probability of 2 successes

48. Chapter 6 – Section 2
For 3 trials, the possible ways of getting exactly 2 successes are
S S F
S F S
F S S
The probabilities for each (using the multiplication rule) are
0.4 • 0.4 • 0.6 = 0.096
0.4 • 0.6 • 0.4 = 0.096
0.6 • 0.4 • 0.4 = 0.096
For 3 trials, the possible ways of getting exactly 2 successes are
S S F
S F S
F S S

49. Chapter 6 – Section 2 The total probability is
But there is a pattern
Each way had the same probability … the probability of 2 success (0.4 times 0.4) times the probability of 1 failure (0.6 times 0.6)
The probability for each case is
0.42 • 0.61
The total probability is
P(2) = = 0.288
The total probability is
P(2) = = 0.288
But there is a pattern
Each way had the same probability … the probability of 2 success (0.4 times 0.4) times the probability of 1 failure (0.6 times 0.6)

50. Chapter 6 – Section 2 There are 3 cases
S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second
S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third
F S S could represent choosing a third combination of 2 out of 3
These are the 3 = 3C2 ways to choose 2 out of 3
There are 3 cases
S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second
S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third
F S S could represent choosing a third combination of 2 out of 3

51. Chapter 6 – Section 2 Thus the total probability
can also be written as
P(2) = 3C2 • .42 • .61
In other words, the probability is
The number of ways of choosing 2 out of 3, times
The probability of 2 successes, times
The probability of 1 failure
Thus the total probability
P(2) = = .288
can also be written as
P(2) = 3C2 • .42 • .61

52. Chapter 6 – Section 2
The general formula for the binomial probabilities is just this
For P(x), the probability of x successes, the probability is
The number of ways of choosing x out of n, times
The probability of x successes, times
The probability of n-x failures
This formula is
P(x) = nCx px (1 – p)n-x
The general formula for the binomial probabilities is just this
The general formula for the binomial probabilities is just this
For P(x), the probability of x successes, the probability is
The number of ways of choosing x out of n, times
The probability of x successes, times
The probability of n-x failures

53. Chapter 6 – Section 2 Example
A student guesses at random on a multiple choice quiz
There are n = 10 questions in total
There are 5 choices per question so that the probability of success p = 1/5 = .2
What is the probability that the student gets 6 questions correct?

54. Chapter 6 – Section 2 Example continued This is a binomial experiment
There are a finite number n = 10 of trials
Each trial has two outcomes (a correct guess and an incorrect guess)
The probability of success is independent from trial to trial (every one is a random guess)
The probability of success p = .2 is the same for each trial

55. Chapter 6 – Section 2 Example continued
The probability of 6 correct guesses is
P(x) = nCx px (1 – p)n-x
= 6C
= 210 • • .4096 =
This is less than a 1% chance
In fact, the chance of getting 6 or more correct (i.e. a passing score) is also less than 1%
Example continued
The probability of 6 correct guesses is
P(x) = nCx px (1 – p)n-x
= 6C
= 210 • • .4096 =
This is less than a 1% chance
Example continued
The probability of 6 correct guesses is
P(x) = nCx px (1 – p)n-x
= 6C
= 210 • • .4096 =
Example continued
The probability of 6 correct guesses is
P(x) = nCx px (1 – p)n-x

56. Chapter 6 – Section 2
Binomial calculations can be difficult because of the large numbers (the nCx) times the small numbers (the px and (1-p)n-x)
It is possible to use tables to look up these probabilities
It is best to use a calculator routine or a software program to compute these probabilities

57. Chapter 6 – Section 2
We would like to find the mean of a binomial distribution
Example
There are 10 questions
The probability of success is .20 on each one
Then the expected number of successes would be 10 • .20 = 2
The general formula
μX = n p

58. Chapter 6 – Section 2
We would like to find the standard deviation and variance of a binomial distribution
This calculation is more difficult
The standard deviation is
σX = √ n p (1 – p)
and the variance is
σX2 = n p (1 – p)

59. Chapter 6 – Section 2 For our random guessing on a quiz problem
x = 6
Therefore
The mean is np = 10 • .2 = 2
The variance is np(1-p) = 10 • .2 • .8 = .16
The standard deviation is √.16 = .4
Remember the empirical rule? A passing grade of 6 is 10 standard deviations from the mean …
For our random guessing on a quiz problem
n = 10
p = .2
x = 6
For our random guessing on a quiz problem
n = 10
p = .2
x = 6
Therefore
The mean is np = 10 • .2 = 2
The variance is np(1-p) = 10 • .2 • .8 = .16
The standard deviation is √.16 = .4

60. Chapter 6 – Section 2
With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution
There are three different shapes for these histograms
When p < .5, the histogram is skewed right
When p = .5, the histogram is symmetric
When p > .5, the histogram is skewed left
With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution

61. Chapter 6 – Section 2 For n = 10 and p = .2 (skewed right) Mean = 2
Standard deviation = .4

62. Chapter 6 – Section 2 For n = 10 and p = .5 (symmetric) Mean = 5
Standard deviation = .5

63. Chapter 6 – Section 2 For n = 10 and p = .8 (skewed left) Mean = 8
Standard deviation = .4

64. Chapter 6 – Section 2
Despite binomial distributions being skewed, the histograms appear more and more bell shaped as n gets larger
This will be important!

65. Summary: Chapter 6 – Section 2
Binomial random variables model a series of independent trials, each of which can be a success or a failure, each of which has the same probability of success
The binomial random variable has mean equal to np and variance equal to np(1-p)

66. Chapter 6 Summary Discrete probability distributions
Random variables with discrete values
Models counts
Binomial distribution
A sequence of success / failure trials
Models the number of successes