Chapter 5 Probability (5.1-5.2)

5.1 definitions and rules Probability is a measure of the likelihood of a random phenomenon. Probability is the long-term proportion with which a certain outcome will occur. Example: Flip a coin 100 times. How many heads do you observe? What is the Probability of observing Heads?

The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.

Tossing a coin n times… The proportion of heads in “n” tosses of a coin changes as we make more tosses. Eventually it approaches 0.5

Question: Hospital A records an average of 50 births a day. Hospital B records an average of 10 births a day. On a particular day, which hospital is more likely to record 80% or more female births? Hospital A (with 50 births a day) Hospital B (with 10 births a day) The two hospitals are equally likely to record such an event. Answer:…?

Definitions Experiment: A planned activity performed to identify its results. Trial: Each repetition of the experiment. Event, E or ei: A result, occurrence, or outcome from a single trial of the experiment. Sample Space, S: A list of all possible events for an experiment.

Probability experiment: having two children. EXAMPLE Identifying Events and Sample Space of a Probability Experiment Probability experiment: having two children. (a) Identify the outcomes of the probability experiment. (b) Determine the sample space. (c) Define the event E = “have one boy”. e1 = boy, boy, e2 = boy, girl, e3 = girl, boy, e4 = girl, girl {(boy, boy), (boy, girl), (girl, boy), (girl, girl)}={e1 , e2 , e3 , e4} {(boy, girl), (girl, boy)} = {e2 , e3}

Tossing a coin: Rolling a single die: MORE EXAMPLES Experiment: Toss a single coin to identify the side shown face up. Trial: A toss of a single coin. Event: “Heads” is shown face up (for example). Sample Space: {Heads, Tails}. Rolling a single die: Experiment: Toss a single die to identify the number of dots shown face up. Trial: A toss of a single die. Event: 3 dots are shown face up (for example). Sample Space: {1, 2, 3, 4, 5, 6}.

All probabilities are between 0 and 1. EXAMPLE In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model. Color Probability Brown 0.12 Yellow 0.15 Red Blue 0.23 Orange Green All probabilities are between 0 and 1. 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1 (the sum of all probabilities must equal 1)

Rules Something that will certainly happen has probability = 1 Can you think of an example? Something that will never happen has probability 0. Example? Probabilities have value between 0 and 1. Probabilities can be expressed as ratios, decimal numbers, or percentages.

Evaluating probabilities

Empirical method: Examples: Records indicate that a river has crested above flood level just four times in the past 2000 years. What is the empirical probability that the river will crest above flood level next year? 4/2000 = 1/500 = 0.002 Number of male births in a given city over the course of a year: In 1987 there were a total of 3,809,394 live births in the U.S., of which 1,951,153 were males. Probability of male birth is: 1,951,153/3,809,394 = 0.5122

theoretical or classical method: (for equally likely outcomes)

EXAMPLE Computing Probabilities Using the Classical Method Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected. (a) What is the probability that it is yellow? (b) What is the probability that it is blue? (c) Comment on the likelihood of the candy being yellow versus blue. There are a total of 9 + 6 + 7 + 4 + 2 + 2 = 30 candies, so N(S) = 30. (b) P(blue) = 2/30 = 0.067. (c) Since P(yellow) = 6/30 and P(blue) = 2/30, selecting a yellow is three times as likely as selecting a blue.

subjective probability: Based on personal judgment or intuition. Value must be between 0 and 1 and be coherent Examples: Probability of finding a parking space downtown on Saturday. An economist predicting there is a 20% chance of recession next year would be a subjective probability.

Exercise: Empirical, Classical, or Subjective? In his fall 1998 article in Chance Magazine, (“A Statistician Reads the Sports Pages,” pp. 17-21,) Hal Stern investigated the probabilities that a particular horse will win a race. He reports that these probabilities are based on the amount of money bet on each horse. When a probability is given that a particular horse will win a race, is this empirical, classical, or subjective probability? Subjective because it is based upon people’s feelings about which horse will win the race. The probability is not based on a probability experiment or counting equally likely outcomes.

5.2 Addition rule Independent, non-overlapping events that have no outcomes in common are called mutually exclusive or disjoint events. The probability of either one or the other happening is: P(E or F) = P(E) + P(F) Example: Suppose you roll a single die. What is the probability of rolling either a 2 or a 3? P(2 or 3) = P(2) + P(3) = 1/6 + 1/6 = 2/6

Venn diagrams representation: The rectangle represents the sample space, and each circle represents an event. Example Let E represent the event “choose a number less than or equal to 2,” and let F represent the event “choose a number greater than or equal to 8.” These events are disjoint as shown in the figure.

EXAMPLE The Addition Rule for Disjoint Events The probability model to the right shows the distribution of the number of rooms in housing units in the United States. Number of Rooms in Housing Unit Probability One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079 Nine or more 0.080 (a) Verify that this is a probability model. All probabilities are between 0 and 1, inclusive. 0.010 + 0.032 + … + 0.080 = 1 Source: American Community Survey, U.S. Census Bureau

Number of Rooms in Housing Unit Probability One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079 Nine or more 0.080 (b) What is the probability a randomly selected housing unit has two or three rooms? P(two or three) = P(two) + P(three) = 0.032 + 0.093 = 0.125

Number of Rooms in Housing Unit Probability One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079 Nine or more 0.080 (c) What is the probability a randomly selected housing unit has one or two or three rooms? P(one or two or three) = P(one) + P(two) + P(three) = 0.010 + 0.032 + 0.093 = 0.135

P(E or F) = P(E) + P(F) - P(E and F) General Addition rule Independent, overlapping events can have outcomes in common. The probability of either one or the other happening is: P(E or F) = P(E) + P(F) - P(E and F) Example: When a computer goes down, there is a probability 0.75 that it is due to an overload and probability 0.15 that it is due to a software problem. There is probability 0.05 that it is due to an overload and a software problem. What is the probability that the computer goes down for an overload or a software problem? Events: E={down for overload} F={down for software problem} P(E)=0.75 P(F)=0.15 P(E and F)=0.05 Result: P(E or F) = P(E) + P(F) - P(E and F)= 0.75+0.15-0.05=0.85

Venn diagrams representation for overlapping events: The rectangle represents the sample space, and each circle represents an event. Example: Let E represent the event “choose a number less than or equal to 3,” and let F represent the event “choose a number greater than or equal to 2.” These events have an overlap as shown in the figure. 0 1 2 3 4 5 6

EXAMPLE Illustrating the General Addition Rule Suppose that a pair of dice are thrown. Let E = “the first die is a two” and let F = “the sum of the dice is less than or equal to 5”. Find P(E or F) using the General Addition Rule.

Complementary events Complement of an Event Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted EC, is all outcomes in the sample space S that are not outcomes in the event E.

Complementary events If there are only two possible outcomes in an uncertain situation, then their probabilities must add to 1. P(EC) = 1 – P(E) Examples: If probability of a single birth resulting in a boy is 0.512, then the probability of it resulting in a girl is 1- 0.512 = 0.488. According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog? P(do not own a dog) = 1 – P(own a dog) = 1 – 0.316 = 0.684

EXAMPLE Computing Probabilities Using Complements The data to the right represent the travel time to work for residents of Hartford County, CT. (a) What is the probability a randomly selected resident has a travel time of 90 or more minutes? There are a total of 24,358 + 39,112 + … + 4,895 = 393,186 residents in Hartford County, CT. The probability a randomly selected resident will have a commute time of “90 or more minutes” is Source: United States Census Bureau

(b) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes. P(less than 90 minutes) = 1 – P(90 minutes or more) = 1 – 0.012 = 0.988 (c) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 15 minutes.