Download Probability and Discrete Probability Distributions

Chapter 4 – Probability and Discrete Probability Distributions
The three types of probability are:
1.
2.
3.


Experiment –
Outcome –
Sample Space –
Examples:
Event –
Examples: Roll a die, Ω = {1, 2, 3, 4, 5, 6}

Let A =

Let B =

Let C =
An event occurs if
Example: If we roll a 4, then which event(s) occurred?
The probability of an event, A, is defined as:
Example: P(A) =
P(C) =
P(B) =
STAT 301 – Autin
1
§ 4.2 – Some Rules of Probability
Rule 1 –
Rule 2 – the complement rule
Example –
Example – What is the probability of drawing an Ace from a standard deck of cards? What is the
probability of drawing something other than an Ace?
Rule 3 – mutually exclusive events
Examples –
Note:
Rule 4 – Additive Rule for mutually exclusive events
Example –
Collectively Exhaustive –
STAT 301 – Autin
2
Rule 5 –
Example –
Rule 6 – Additive Rule
Example – Find the probability of drawing a red card or an Ace.
Rule 7 – Conditional Probability –
Example – Draw a single card from a standard deck. What is the probability that the card is a diamond,
given that it is red?
Example – Roll a die. What is the probability that you get a 2, given that you got an even number?
Independent Events –
STAT 301 – Autin
3
Examples –
Example – A single card is drawn from a standard deck. Are drawing a face card and drawing a red
card independent events?
Rule 8 – Multiplicative Rule for independent events
Example – Find the probability of drawing two Queens from a deck of cards if it is done with
replacement.
Rule 9 – Multiplicative Rule
Example – Find the probability of drawing two Queens from a deck of cards if it is done without
replacement.
Example: DVDs of the top 50 grossing movies are located in a bin. The movies are categorized by
rating and decade in the contingency table below.
STAT 301 – Autin
G
PG
PG-13
R
2000s
2
13
22
2
1990s
1
1
7
0
1980s
0
1
0
0
1970s
0
1
0
0
4
Based on the data in the contingency table, find the following probabilities.

P(1990s)

P(1990s and PG-13)

P(PG-13)
Are 1990s and PG-13 mutually exclusive events?

The probability that a randomly selected DVD is rated PG-13 or is from the 1990s.

The probability that a randomly selected DVD is from the 1990s given that it rated PG-13.

Are PG-13 and 1990s independent or dependent events?
§ 4.3 – The Probability Distribution
Probability Distribution –

Example – The number of training units that must be passed before a complex computer software
program is mastered varies from one to five, depending on the student. The software manufacturer has
determined the following probability distribution for the number of training units required to master the
software.
X
1
2
3
4
5
P(X)
0.1
0.25
0.4
0.15
0.1
STAT 301 – Autin
5
Expected Value –
Example: Find the mean number of training units required to master the software.
Variance –
Standard Deviation –
Example: Find the standard deviation of the number of training units required to master the software.
§ 4.4 – The Binomial Distribution
Many experiments result in dichotomous responses, where there are only two possible outcomes:
Examples: Yes—No
True—False
Defective—Nondefective
Alive—Dead
Suppose that for each run of the experiment, the probability of a certain outcome stays the same. Now suppose
that you are interested in counting the number of times a particular outcome occurs.
Example: Suppose you are playing a game in which you win 25¢ for each time a coin flip lands tails. You are
allowed to flip a fair coin 10 times.
Characteristics of a Binomial Distribution:
1.
2.
3.
STAT 301 – Autin
6
The random variable X is defined as:
Two parameters of the binomial distribution:
The binomial probability distribution:
Example: Find the probability of rolling six 1’s in ten rolls of a die.
Finding binomial probabilities using a TI-83/84:
Press to display the DISTR menu. Press the down arrow until binompdf( is highlighted and press ENTER.
Then enter “n, p, x)” where n is the number of trials, p is the probability of success, and x is the value of interest.
Examples:
Finding binomial probabilities using a TI-89:
If you do not already have it, you must first download and install the Statistics with List Editor Application from
http://education.ti.com/educationportal/sites/US/productDetail/us_statslist_89.html (or see me).
Select the
Stats/List Editor from the main menu. Press  to select the Distr menu. Press the down arrow until B:
Binomial Pdf is highlighted and press ENTER. Enter n, p, and x in the appropriate lines, and hit ENTER. The
answer will be displayed.
Example:
STAT 301 – Autin
7
Example: In the die example, suppose you wanted to find the probability that you roll at most six 1’s. How
would you do this?
Cumulative binomial probability:
Finding cumulative binomial probabilities using a TI-83/84:
Press to display the DISTR DRAW menu. Press the down arrow until binomcdf( is highlighted and press
ENTER. Then enter “n, p, x)” where n is the number of trials, p is the probability of success, and x is the upper
limit for your cumulative probability.
Example:
Finding binomial cumulative probabilities using a TI-89:
Select the Stats/List Editor from the main menu. Press  to select the Distr menu. Press the down arrow until
C: Binomial Cdf is highlighted and press ENTER. Enter n, p, and the lower (0) and upper limit values for x in
the appropriate lines, and hit ENTER. The answer will be displayed.
Example:
Example: What is the probability of rolling at most four 1’s on 10 rolls of a die?
Example: What is the probability of rolling more than six 1’s on 10 rolls of a die?
Example: What is the probability of rolling fewer than six 1’s on 10 rolls of a die?
Example: What is the probability of rolling two to five 1’s on 10 rolls of a die?
STAT 301 – Autin
8
Shape of the Binomial Distribution
If p = 0.5:
If p < 0.5:
If p > 0.5:
Mean (Expected Value) of the Binomial Distribution
Example: Find the mean number of 1’s rolled in 10 rolls of a die.
Standard Deviation of the Binomial Distribution
Example: Find the standard deviation for the number of 1’s rolled in 10 rolls of a die.
Example: A manufacturer of CD players subjects the equipment to a comprehensive testing process for all
mechanical and electrical functions before the equipment leaves the factory. Ideally, the hope is that each CD
player passes on the first test. Suppose that past data indicates that 90% of CD players pass the first test.
Suppose twenty CD players are randomly selected for inspection.
 Define the random variable of interest, the number of trials, and the probability of success.

What is the probability that all of these CD players pass the first test?

What is the probability that fewer than 15 of these CD players pass the first test?

What is the probability that at least 15 of these CD players pass the first test?
STAT 301 – Autin
9

What is the probability that 15 to 18 of these CD players pass the first test?

What is the probability that 3 of these CD players fail the first test?

What is the expected number of CD players that will pass the first test?

What is the standard deviation for the number of CD players that will pass the first test?
Combination notation –
Finding combinations using a TI-83/84:
When finding  n  , first type the value of n. Then press MATH. Arrow over to the PRB menu, and choose 3:
x
 
nCr. Now type the value of x, and press ENTER.
Example: 
10 
 6   210
 
Finding combinations using a TI-89:
Press MATH. Arrow down to 7: Probability. Choose 3: nCr(. Type “n, x)” where n and x are the appropriate
values, and press ENTER.
Example:
STAT 301 – Autin
10
§ 4.5 – The Hypergeometric Distribution
Suppose that the probability of a success on the outcome of a particular trial is not the same as on a different
trial because the data have been selected without replacement from a finite population. Since the trials are not
identical and independent, this is no longer a binomial random variable.
Hypergeometric distribution –
Example: Suppose that a standard deck of cards is shuffled and five cards are to be selected without
replacement.
 What is the probability that four of the cards will be red?

What is the probability that two of the cards will be face cards?
Mean (Expected Value) of the Hypergeometric Distribution
Example: What is the average number of face cards that you would expect when drawing five cards
from a deck without replacement?
STAT 301 – Autin
11
Standard Deviation of the Hypergeometric Distribution
Example: What is the standard deviation of the number of face cards when drawing five cards from a
deck without replacement?
§ 4.6 – The Negative Binomial and Geometric Distributions
Suppose that instead of being interested in the number of successes, we want to know when a certain success
will occur.
Negative Binomial Distribution –
Example: Suppose that seven in twenty computer scientists have worked on a neural-net program. A
representative for a company who is seeking computer scientists who have worked on a neural-net program will
continue to interview computer scientists until she finds two who have worked on a neural-net program. What is
the probability that the interviewing process will end with the fifth interview?
Mean of the Negative Binomial Distribution
Example: What is the expected number of interviews that need to be done until she finds two computer
scientists who have worked on a neural-net program?
STAT 301 – Autin
12
Standard Deviation of the Negative Binomial Distribution
Example: What is the standard deviation for the number of interviews that need to be done until she
finds two computer scientists who have worked on a neural-net program?
Suppose we are interested in knowing when the first success occurs. This is a special case of the negative
binomial distribution called the geometric distribution.
Geometric Distribution –
Example: 5% of children are born with overlapping toes (usually with no harmful effect). If a doctor searches
the records of births at a particular hospital, what is the probability that the first record showing a child born with
overlapping toes occurs on the tenth record searched?
Calculator Note: Geometric probabilities and cumulative probabilities can also be found using a graphing
calculator. The functions (geometpdf and geometcdf on the TI-83/84 and Geometric Pdf and Geometric Cdf on
the TI-89) are found in the same menus as the binomial functions. You must give the functions the values of p
and n (in that order, separated by commas, for the TI-83/84).
Example: Use a calculator to find the probability that the first record showing a child born with overlapping toes
occurs on the fifth record searched.
Example: What is the probability that the first record showing a child born with overlapping toes occurs in at
most ten records?
Mean of the Geometric Distribution
Example: Find the mean number of records that must be searched until the first one with overlapping
toes occurs.
STAT 301 – Autin
13
Standard Deviation of the Geometric Distribution
Example: Find the standard deviation of the number of records that must be searched until the first one
with overlapping toes occurs.
§ 4.6 – The Poisson Distribution
A probability distribution that is often useful in describing the number of events that will occur in a specific period
of time or in a specific area or volume is the Poisson distribution.
Examples of Poisson random variables:
Characteristics of a Poisson Distribution:
1.
Consider dividing the area of opportunity into very small, equal-sized subareas. Then
2.
3.
4.
The random variable X is defined as:
Parameter of the Poisson distribution:
STAT 301 – Autin
14
The Poisson probability distribution:
Example: U.S. airlines average about 3.8 fatalities per month. Assume that the probability distribution for the
number of airline fatalities per month can be approximated by a Poisson distribution. What is the probability that
one airline fatality will occur during any given month.
Calculator Note: Poisson probabilities and cumulative probabilities can also be found using a graphing
calculator. The functions (poissonpdf and poissoncdf on the TI-83/84 and Poisson Pdf and Poisson Cdf on the
TI-89) are found in the same menus as the binomial functions. You must give the functions the values of λ and
x (in that order, separated by commas, for the TI-83/84).
Example: Use a calculator to find the probability that two airline fatalities will occur during any given month.
Example: What is the probability that more than 3 airline fatalities will occur during any given month?
Mean of the Poisson Distribution
Example: What is the expected number of airline fatalities in any given month?
Standard Deviation of the Poisson Distribution
Example: What is the standard deviation for the number of airline fatalities in a month?
STAT 301 – Autin
15