A Mathematical View of Our World

A Mathematical View of Our World 1st ed. Parks, Musser, Trimpe, Maurer, and Maurer

Chapter 10 Probability

Section 10.1Simple Experiments • Goals • Study probability • Experimental probability • Theoretical probability • Study probability properties • Mutually exclusive events • Unions and intersections of events • Complements of events

10.1 Initial Problem • Three cards were removed from wedding presents and then randomly replaced. • What are the chances that at least one of the gifts was paired with the correct card? • The solution will be given at the end of the section.

Interpreting Probability • Probability is the mathematics of chance. • For example, the statement “The chances of winning the lottery game are 1 in 150,000” means that only 1 of every 150,000 lottery tickets printed is a winning ticket.

Probability Terminology • Making an observation or taking a measurement is called an experiment. • An outcome is one of the possible results of an experiment. • The set of all possible outcomes is called the sample space. • An event is any collection of possible outcomes.

Example 1 • The experiment consists of rolling a standard six-sided die and recording the number of dots showing on the top face. • List the sample space. • List one possible event.

Example 1, cont’d • Solution: The sample space contains 6 possible outcomes and can be written {1, 2, 3, 4, 5, 6}. • One possible event is {2, 4, 6}, which is the event of getting an even number of dots.

Example 2 • The experiment consists of tossing a coin 3 times and recording the results in order. • List the sample space. • List one possible event.

Example 2, cont’d • Solution: The sample space contains 8 possible outcomes and can be written {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. • One possible event is {HTH, HTT, TTH, TTT}, which is the event of getting a tail on the second coin toss.

Example 3 • The experiment consists of spinning a spinner twice and recording the colors it lands on. • List the sample space. • List one possible event.

Example 3, cont’d • Solution: The sample space contains 16 possible outcomes and can be written {RR, RY, RG, RB, YR, YY, YG, YB, GR, GY, GG, GB, BR, BY, BG, BB}. • One possible event is {RR, YY, GG, BB}, which is the event of getting the same color on both spins.

Example 4 • The experiment consists of rolling 2 standard dice and recording the number appearing on each die. • List the sample space. • List one possible event.

Example 4, cont’d • Solution: The sample space contains 36 possible outcomes and can be written {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.

Example 4, cont’d • Solution, cont’d: One possible event is {(6,1), (5,2), (4,3), (3,4), (2,5), (1,6)}, which is the event of getting a total of 7 dots on the two dice.

Question: 1. An experiment consists of tossing a coin and then rolling a 4-sided die. List the outcomes in the sample space a. { H1, H2, H3, H4, T1, T2, T3, T4 } b. { H, T, 1, 2, 3, 4 } c. { H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 } d. { H, T, 1, 2, 3, 4, 5, 6 }

Probability, cont’d • The probability of an event is a number from 0 to 1, and can be written as a fraction, decimal, or percent. • The greater the probability, the more likely the event is to occur. • An impossible event has probability 0. • A certain event has probability 1.

Experimental Probability • One way to find the probability of an event is to conduct a series of experiments. • The experimental probability is the relative frequency with which an event occurs in a particular sequence of trials.

Example 5 • An experiment consisted of tossing 2 coins 500 times and recording the results • Let E be the event of getting a head on the first coin and find the experimental probability of E.

Example 5, cont’d • Solution: The event E is {HH, HT}. • Event E occurred a total of 137 + 115 = 252 times out of 500. • The experimental probability of E is

Question: A total of 200 people are given a taste test of 2 kinds of crackers. The results are that 148 of them prefer Cracker A, 41 of them prefer Cracker B, and 11 have no preference. Find the experimental probability of a randomly selected person preferring Cracker B. a. 74.0% b. 5.5% c. 20.5% d. 27.7%

Theoretical Probability • Another way to find the probability of an event is to use the theory of what “should” happen rather than conducting experiments. • The theoretical probability is the chance an event will occur based on the situation, such as tossing a fair coin and knowing each side should come up half of the time.

Theoretical Probability, cont’d • If all the outcomes in a sample space are equally likely to occur, then the probability of event E is equal to the number of outcomes in E divided by the number of outcomes in the sample space S. • The probability of event E is written P(E).

Example 6 • An experiment consists of tossing 2 fair coins. • Find the theoretical probability of: • Each outcome in the sample space. • The event E of getting a head on the first coin. • The event of getting at least one head.

Example 6, cont’d • Solution: • There are 4 outcomes in the sample space: {HH, HT, TH, TT}. Each outcome is equally likely to occur.

Example 6, cont’d • Solution, cont’d: • The event E is {HH, HT} and the theoretical probability of E is the number of outcomes in E divided by the number of outcomes in the sample space.

Example 6, cont’d • Solution, cont’d: • The event of getting at least one head is E = {HH, HT, TH}.

Example 7 • An experiment consists of rolling 2 fair dice. • Find the theoretical probability of: • Event A: getting 7 dots. • Event B: getting 8 dots. • Event C: getting at least 4 dots.

Example 7, cont’d • Solution: There are 36 outcomes in the sample space. • The event A contains 6 outcomes: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. Each outcome is equally likely to occur.

Example 7, cont’d • Solution, cont’d: • The event B contains 5 equally likely outcomes: {(2,6), (3,5), (4,4), (5,3), (6,2)}.

Example 7, cont’d • Solution, cont’d: • The event C contains 33 equally likely outcomes: {(1,3), (1,4), (1,5), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.

Example 8 • A jar contains four marbles: 1 red, 1 green, 1 yellow, and 1 white.

Example 8, cont’d • If we draw 2 marbles in a row, without replacing the first one, find the probability of: • Event A: One of the marbles is red. • Event B: The first marble is red or yellow. • Event C: The marbles are the same color. • Event D: The first marble is not white. • Event E: Neither marble is blue.

Example 8, cont’d • Solution: The sample space contains 12 outcomes: {RG, RY, RW, GR, GY, GW, YR, YG, YW, WR, WG, WY}. • Event A: One of the marbles is red. • A = {RG, RY, RW, GR, YR, WR}.

Example 8, cont’d • Solution, cont’d: • Event B: The first marble is red or yellow. • B = {RG, RY, RW, YR, YG, YW}. • Event C: The marbles are the same color. • C = { }.

Example 8, cont’d • Solution, cont’d: • Event D: The first marble is not white. • D = {RG, RY, RW, GR, GY, GW, YR, YG, YW}. • Event E: Neither marble is blue. • E = S

Question: A jar contains 1 red marble, 1 green marble, and 1 blue marble. You draw 2 marbles in a row, without replacing the first one. What is the probability of the event E of the first marble being red or yellow? a. P(E) = 0 b. P(E) = 1/3 c. P(E) = ½ d. P(E) = 2/3

Union and Intersection • The union of two events, A U B, refers to all outcomes that are in one, the other, or both events. • The intersection of two events, A∩ B, refers to outcomes that are in both events.

Mutually Exclusive Events • Events that have no outcomes in common are said to be mutually exclusive. • If A and B are mutually exclusive events, then

Example 9 • A card is drawn from a standard deck of cards. • Let A be the event the card is a face card. • Let B be the event the card is a black 5. • Find and interpret P(A U B).

Example 9, cont’d • Solution: The sample space contains 52 equally likely outcomes.

Example 9, cont’d • Solution, cont’d: Event A has 12 outcomes, one for each of the 3 face cards in each of the 4 suits. • P(A) = 12/52. • Event B has 2 outcomes, because there are 2 black fives. • P(B) = 2/52.

Example 9, cont’d • Solution, cont’d: Events A and B are mutually exclusive because it is impossible for a 5 to be a face card. • P(A U B) = 12/52 + 2/52 = 14/52 = 7/26. • This is the probability of drawing either a face card or a black 5.

Complement of an Event • The set of outcomes in a sample space S, but not in an event E, is called the complement of the event E. • The complement of E is written Ē.

Complement of an Event, cont’d • The relationship between the probability of an event E and the probability of its complement Ē is given by:

Example 10 • In a number matching game, • First Carolan chooses a whole number from 1 to 4. • Then Mary guesses a number from 1 to 4. • What is the probability the numbers are equal? • What is the probability the numbers are unequal?

Example 10, cont’d • Solution: The sample space contains 16 outcomes: { (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) }. • Let E be the event the numbers are equal. • P(E) = ¼ • Then Ēis the event the numbers are unequal. • P(Ē) = 1 – ¼ = ¾

Example 11 • A diagram of a sample space S for an experiment with equally likely outcomes is shown.

Example 11, cont’d • Find the probability of each of the events: • S • A • B • C

Example 11, cont’d • Solution:

A Mathematical View of Our World