Chapter 5:Chapter 5:ProbabilityProbability

STP 226: Elements of StatisticsSTP 226: Elements of Statistics

Jenifer BoshesJenifer Boshes

Arizona State UniversityArizona State University

Example 1:Example 1:

Recall the line of succession for the Presidency: Recall the line of succession for the Presidency: Vice-President (V), Speaker of the House (H), President pro tempore of the Senate (P), Secretary of State (S), Secretary of the Treasury (T).

1. How many samples of size 2 are there from this “population”?2. What is the probability that the Vice-President and Secretary of State are

chosen? 3. What is the probability that the Vice President is in the randomly chosen

sample? How do we know?

In general:

N

f

outcomes possible of # total

occurcan event an waysof # Probability of an event

Experiments & EventsExperiments & Events

An An experimentexperiment is an action whose is an action whose outcome cannot be predicted with outcome cannot be predicted with certainty.certainty.

An An eventevent is a specified result that may or is a specified result that may or may not occur when an experiment is may not occur when an experiment is performed.performed.

Example 2:Example 2:On October 4th, 2008, Congress passed On October 4th, 2008, Congress passed

HR 1424, more commonly known as HR 1424, more commonly known as the “Bailout Bill”. The table that follows the “Bailout Bill”. The table that follows shows the number of Congressmen shows the number of Congressmen who voted for the bill, categorized by who voted for the bill, categorized by astrological sign.astrological sign.

A congressman is selected at random.

• What is the probability he or she is a Leo?

• What is the probability he or she was born between June 22 and September 23?

• What is the probability he or she was born after October 24?

Dates FreqAquarius (January 20 – February 18) 20

Pisces (February 19 – March 20) 12Aries (March 21 – April 20) 20Taurus (April 20 – May 21) 18Gemini (May 22 – June 21) 24Cancer (June 22 – July 23) Leo (July 24 – August 23)

Virgo (August 24 – September 23)Libra (September 24 – October 23) Scorpio (October 24 – November 22)

Sagittarius (November 23 – December 21)Capricorn (December 22 – January 19)

31232924222020

TOTAL: 263

Sample SpaceSample Space

The The sample spacesample space is the collection of all is the collection of all possible outcomes for an experiment.possible outcomes for an experiment.

Example 3:Example 3:Write the sample space for tossing two balanced Write the sample space for tossing two balanced

dice.dice.

S = {S = { (1,1),(1,1), (2,1),(2,1), (3,1),(3,1), (4,1),(4,1), (5,1),(5,1), (6,1),(6,1),

(1,2),(1,2), (2,2),(2,2), (3,2),(3,2), (4,2),(4,2), (5,2),(5,2), (6,2),(6,2),

(1,3),(1,3), (2,3),(2,3), (3,3),(3,3), (4,3),(4,3), (5,3),(5,3), (6,3),(6,3),

(1,4),(1,4), (2,4),(2,4), (3,4),(3,4), (4,4),(4,4), (5,4),(5,4), (6,4),(6,4),

(1,5),(1,5), (2,5),(2,5), (3,5),(3,5), (4,5),(4,5), (5,5),(5,5), (6,5),(6,5),

(1,6),(1,6), (2,6),(2,6), (3,6),(3,6), (4,6),(4,6), (5,6),(5,6), (6,6) }(6,6) }

Example 3:Example 3:

Example 4:Example 4:

Two balanced dice are rolled. What is the Two balanced dice are rolled. What is the probability that:probability that:

(a)(a) the sum of the dice is 11?the sum of the dice is 11?

(b)(b) doubles are rolled?doubles are rolled?

Basic Properties of ProbabilityBasic Properties of Probability

(1)(1) The probability of an event is always The probability of an event is always between 0 and 1, inclusive. (Why?)between 0 and 1, inclusive. (Why?)

(2)(2) The probability of an event that cannot The probability of an event that cannot occur is 0. (Impossible event.)occur is 0. (Impossible event.)

(3)(3) The probability of an event that must The probability of an event that must occur is 1. (Certain event.)occur is 1. (Certain event.)

Example 5:Example 5:

Two balanced dice are rolled. What is the Two balanced dice are rolled. What is the probability that:probability that:

(a)(a) the sum of the dice is 10 or more?the sum of the dice is 10 or more?

(b)(b) the sum of the dice is 1?the sum of the dice is 1?

(c)(c) the sum of the dice is 1 or more?the sum of the dice is 1 or more?

Frequentist Interpretation of Frequentist Interpretation of ProbabilityProbability

The probability of an event is to be the proportion of times it occurs in a large number of repetitions of an experiment.

Standard Deck of 52 CardsStandard Deck of 52 Cards

13 “Ranks” in Each Suit13 “Ranks” in Each Suit2 3 4 5 6 7 8 9 10 J 2 3 4 5 6 7 8 9 10 J (Jack)(Jack) Q Q (Queen)(Queen) K K (King)(King) A A (Ace)(Ace)

*The Jack, Queen, and King are commonly referred to as “face cards.”*The Jack, Queen, and King are commonly referred to as “face cards.”

52 Cards52 Cards

4 Suits (Clubs, Diamonds, Hearts, Spades)4 Suits (Clubs, Diamonds, Hearts, Spades)

Standard Deck of 52 CardsStandard Deck of 52 Cards

22 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 J J Q Q K K A A22 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 J J Q Q K K A A

22 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 J J Q Q K K A A22 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 J J Q Q K K A A

Probability NotationProbability Notation

• If E is an event, then P(E) is the probability that event E will occur.

Example 6:Example 6:

Consider the experiment of selecting one card from a standard deck of 52.D = event the card selected is a diamond F = event the card selected is a face cardE = event the card selected is a king G = event the card selected is a 10 or an ace

Find:

(a) P(D)

(b) P(E)

(c) P(F)

(d) P(G)

Complementation RuleComplementation Rule• Every event E has a corresponding event defined by the condition that “E does not

occur.” This event is called the complement of E and is notated by .E

Example 6:Example 6:

Consider the experiment of selecting one card from a standard deck of 52.D = event the card selected is a diamond F = event the card selected is a face cardE = event the card selected is a king G = event the card selected is a 10 or an aceFind:

(g) P(D or F)

(h) P(D and F)

)( (e) EP)( (f) FP

Example 6:Example 6:

Consider the experiment of selecting one card from a standard deck of 52.D = event the card selected is a diamond F = event the card selected is a face cardE = event the card selected is a king G = event the card selected is a 10 or an ace

Find:

(i) P(D or E)

)or ( (j) EDP

Example 7:Example 7:In craps, a player rolls two balanced dice. Let A = event the sum of the faces is 7B = event the sum of the faces is 11 C = event the sum of the faces is 2D = event the sum of the faces is 3 E = event the sum of the faces is 12F = event the sum of the faces is 10 G = event doubles are rolled

(a)Find the probability of each event.(b)The player wins on the first roll if the sum is 7 or 11. Find the probability of this event.(c)The player loses on the first roll if the sum is 2, 3, or 12. Find the probability of this event.(d)Find P(E and G). (e)Find P(F or G).

.or Find (f) GFP

BibliographyBibliography

Some of the textbook images embedded in the slides Some of the textbook images embedded in the slides were taken from:were taken from:

Elementary StatisticsElementary Statistics, Sixth Edition; by Weiss; Addison , Sixth Edition; by Weiss; Addison Wesley Publishing Company Wesley Publishing Company

Copyright © 2005, Pearson Education, Inc.Copyright © 2005, Pearson Education, Inc.