Chapter 5 Probability STP 226 Elements of Statistics

Chapter 5: Probability STP 226: Elements of Statistics Jenifer Boshes Arizona State University

Example 1: 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. 2. 3. How many samples of size 2 are there from this “population”? What is the probability that the Vice-President and Secretary of State are chosen? What is the probability that the Vice President is in the randomly chosen sample? How do we know? In general: Probability of an event

Experiments & Events An experiment is an action whose outcome cannot be predicted with certainty. An event is a specified result that may or may not occur when an experiment is performed.

Example 2: On October 4 th, 2008, Congress passed HR 1424, more commonly known as the “Bailout Bill”. The table that follows shows the number of Congressmen who voted for the bill, categorized by 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 Freq Aquarius (January 20 – February 18) 20 Pisces (February 19 – March 20) 12 Aries (March 21 – April 20) 20 Taurus (April 20 – May 21) 18 Gemini (May 22 – June 21) 24 Cancer (June 22 – July 23) 31 Leo (July 24 – August 23) 23 Virgo (August 24 – September 23) 29 Libra (September 24 – October 23) 24 Scorpio (October 24 – November 22) 22 Sagittarius (November 23 – December 21) 20 Capricorn (December 22 – January 19) 20 TOTAL: 263

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

Example 3: Write the sample space for tossing two balanced dice. S = { (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6) }

Example 3:

Example 4: Two balanced dice are rolled. What is the probability that: (a) the sum of the dice is 11? (b) doubles are rolled?

Basic Properties of Probability (1) The probability of an event is always between 0 and 1, inclusive. (Why? ) (2) The probability of an event that cannot occur is 0. (Impossible event. ) (3) The probability of an event that must occur is 1. (Certain event. )

Example 5: Two balanced dice are rolled. What is the probability that: (a) the sum of the dice is 10 or more? (b) the sum of the dice is 1? (c) the sum of the dice is 1 or more?

Frequentist Interpretation of Probability 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 Cards 4 Suits (Clubs, Diamonds, Hearts, Spades) 13 “Ranks” in Each Suit 2 3 4 5 6 7 8 9 10 J (Jack) Q (Queen) K (King) A (Ace) *The Jack, Queen, and King are commonly referred to as “face cards. ”

Standard Deck of 52 Cards 2 3 4 5 6 7 8 9 10 J Q K A

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

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 card E = 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 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.

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 card E = event the card selected is a king G = event the card selected is a 10 or an ace Find: (g) P(D or F) (h) P(D and F)

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 card E = event the card selected is a king G = event the card selected is a 10 or an ace Find: (i) P(D or E)

Example 7: In craps, a player rolls two balanced dice. Let A = event the sum of the faces is 7 B = event the sum of the faces is 11 C = event the sum of the faces is 2 D = event the sum of the faces is 3 E = event the sum of the faces is 12 F = event the sum of the faces is 10 G = event doubles are rolled (a) (b) (c) (d) (e) Find the probability of each event. The player wins on the first roll if the sum is 7 or 11. Find the probability of this event. The player loses on the first roll if the sum is 2, 3, or 12. Find the probability of this event. Find P(E and G). Find P(F or G).

Bibliography Some of the textbook images embedded in the slides were taken from: Elementary Statistics, Sixth Edition; by Weiss; Addison Wesley Publishing Company Copyright © 2005, Pearson Education, Inc.