5.1 Probability of Simple Events Chapter 5 Probability 5.1 Probability of Simple Events

Probability:

Probability: Probability is the branch of mathematics which studies “randomness.”

Probability: Probability is the branch of mathematics which studies “randomness.” Probability was first formally developed in the middle of the 17th century by the French mathematicians Fermat and Pascal.

Probability: Probability is the branch of mathematics which studies “randomness.” Probability was first formally developed in the middle of the 17th century by the French mathematicians Fermat and Pascal. Chevalier De Mere asked Pascal to determine the chances of winning various games of chance (dice, cards, and roulette.)

Probability was developed from these beginnings over the 18th and 19th centuries.

Probability was developed from these beginnings over the 18th and 19th centuries. The mathematical foundations of the subject were formalized by Kolmogorov and other Soviet mathematicians in the middle of the 20th century.

Probability was developed from these beginnings over the 18th and 19th centuries. The mathematical foundations of the subject were formalized by Kolmogorov and other Soviet mathematicians in the middle of the 20th century. Probability provides the theoretical basis for statistical practice.

Why use probability?

Why use probability? Many events can’t be “predicted” before they happen.

Why use probability? Many events can’t be “predicted” before they happen. That is, we don’t know what the outcome of the event will be before it happens.

Many events can’t be “predicted” before they happen. Why use probability? Many events can’t be “predicted” before they happen. That is, we don’t know what the outcome of the event will be before it happens. Will a given person develop lung cancer in the course of his lifetime?

Many events can’t be “predicted” before they happen. Why use probability? Many events can’t be “predicted” before they happen. That is, we don’t know what the outcome of the event will be before it happens. Will a given person develop lung cancer in the course of his lifetime? Will this person develop lung cancer if he smokes three packs a day?

This last example points out how the study of “randomness” might be a useful thing.

This last example points out how the study of “randomness” might be a useful thing. Just because an event is random doesn’t mean we can’t make some type of judgment about the “chances” of it occurring.

This last example points out how the study of “randomness” might be a useful thing. Just because an event is random doesn’t mean we can’t make some type of judgment about the “chances” of it occurring. The chances someone who does not smoke will develop lung cancer in the course of his lifetime is about 1 in 200.

This last example points out how the study of “randomness” might be a useful thing. Just because an event is random doesn’t mean we can’t make some type of judgment about the “chances” of it occurring. The chances someone who does not smoke will develop lung cancer in the course of his lifetime is about 1 in 200. The chances someone who smokes will develop lung cancer is 14 in 200.

Example: There are 30 MLB teams

Example: There are 30 MLB teams Number the teams from 1 to 30.

Example: There are 30 MLB teams Number the teams from 1 to 30. At the start of the season a number between 1 and 30 is randomly selected.

Example: There are 30 MLB teams Number the teams from 1 to 30. At the start of the season a number between 1 and 30 is randomly selected. What is the probability that the team that has been selected wins the World Series?

Example: There are 30 MLB teams Number the teams from 1 to 30. At the start of the season a number between 1 and 30 is randomly selected. What is the probability that the team that has been selected wins the World Series? What is the probability if you learn that the team that has been selected is the NY Yankees?

Thus even though an event is random, this does not mean we can’t say anything about it.

Thus even though an event is random, this does not mean we can’t say anything about it. We can make judgments about how likely or unlikely is it to happen.

We can make judgments about how likely or unlikely is it to happen. Thus even though an event is random, this does not mean we can’t say anything about it. We can make judgments about how likely or unlikely is it to happen. This is what sports odds-makers do when they give odds for teams to win or lose.

We can make judgments about how likely or unlikely is it to happen. Thus even though an event is random, this does not mean we can’t say anything about it. We can make judgments about how likely or unlikely is it to happen. This is what sports odds-makers do when they give odds for teams to win or lose. The more information we have about a situation the better we can judge how likely something is to happen.

Example: What is the safest form of travel?

Example: What is the safest form of travel? Auto

Example: What is the safest form of travel? Auto Train

Example: What is the safest form of travel? Auto Train Bus

Example: What is the safest form of travel? Auto Train Bus Airplane

Example: What is the safest form of travel? Deaths per 100 million miles Auto Train Bus Airplane

Example: What is the safest form of travel? Deaths per 100 million miles Auto ,94 Train Bus Airplane

Example: What is the safest form of travel? Deaths per 100 million miles Auto ,94 Train .04 Bus Airplane

Example: What is the safest form of travel? Deaths per 100 million miles Auto ,94 Train .04 Bus .02 Airplane

Example: What is the safest form of travel? Deaths per 100 million miles Auto ,94 Train .04 Bus .02 Airplane .01

Example: What is the safest form of travel? Deaths per 100 million miles Auto ,94 Train .04 Bus .02 Airplane .01 You are 94 times more likely to die from an auto accident than from a plane crash.

Example: What is the safest form of travel? Deaths per 100 million miles Auto ,94 Train .04 Bus .02 Airplane .01 You are 94 times more likely to die from an auto accident than from a plane crash. We can adjust our actions based on how likely an event is given a certain course of action.

Research studies involving statistics never determines anything with absolute 100% “certainty.”

Research studies involving statistics never determines anything with absolute 100% “certainty.” What these studies attempt to do is determine the probability with which certain events may or may not occur.

Research studies involving statistics never determines anything with absolute 100% “certainty.” What these studies attempt to do is determine the probability with which certain events may or may not occur. Research often involves determining the effect of some explanatory variable on the probability that some outcome variable will happen.

Example:

Example: Smoking increases the probability of developing lung cancer by a factor of 14.

Example: Smoking increases the probability of developing lung cancer by a factor of 14. Saying “smoking causes lung cancer” is misleading: the vast majority of smokers will never develop lung cancer.

Example: Smoking increases the probability of developing lung cancer by a factor of 14. Saying “smoking causes lung cancer” is misleading: the vast majority of smokers will never develop lung cancer. However, there is a very strong connection between smoking and lung cancer. Roughly 90% of all people with lung cancer are smokers.

Example: Smoking increases the probability of developing lung cancer by a factor of 14. Saying “smoking causes lung cancer” is misleading: the vast majority of smokers will never develop lung cancer. However, there is a very strong connection between smoking and lung cancer. Roughly 90% of all people with lung cancer are smokers. Smoking greatly increases the probability of getting lung cancer.

(Mathematical) Probability is a numerical measure of the likelihood of a random event.

(Mathematical) Probability is a numerical measure of the likelihood of a random event. For one instance of a given random event we cannot be sure if it is going to happen or not.

(Mathematical) Probability is a numerical measure of the likelihood of a random event. For one instance of a given random event we cannot be sure if it is going to happen or not. However, probability gives the long-term or large sample proportion with which certain outcomes will occur.

One Definition of Probability:

One Definition of Probability: Perform an experiment and observe a random event.

One Definition of Probability: Perform an experiment and observe a random event. Either a certain outcome occurs or it does not occur.

One Definition of Probability: Perform an experiment and observe a random event. Either a certain outcome occurs or it does not occur. We keep repeating the experiment over and over again and calculate the percentage of times out of all the experiments that this event occurs.

One Definition of Probability: Perform an experiment and observe a random event. Either a certain outcome occurs or it does not occur. We keep repeating the experiment over and over again and calculate the percentage of times out of all the experiments that this event occurs. The long-term proportion (%) with which a certain outcome is observed is the probability of that outcome.

Example:

Example: Experiment: flip a quarter.

Example: Experiment: flip a quarter. Event: determine whether the coin lands “heads.”

Example: Experiment: flip a quarter. Event: determine whether the coin lands “heads.” Suppose I perform this “experiment” 10 times and get the following outcomes:

Example: Experiment: flip a quarter. Event: determine whether the coin lands “heads.” Suppose I perform this “experiment” 10 times and get the following outcomes: H T T H H T T H T T.

Example: Experiment: flip a quarter. Event: determine whether the coin lands “heads.” Suppose I perform this “experiment” 10 times and get the following outcomes: H T T H H T T H T T. The proportion of events which are heads is 4/10 = 0.4.

Example: Experiment: flip a quarter. Event: determine whether the coin lands “heads.” Suppose I perform this “experiment” 10 times and get the following outcomes: H T T H H T T H T T. The proportion of events which are heads is 4/10 = 0.4. For any one flip of the coin, I cannot be sure whether it will show up heads or tails.

Example: Experiment: flip a quarter. Event: determine whether the coin lands “heads.” Suppose I perform this “experiment” 10 times and get the following outcomes: H T T H H T T H T T. The proportion of events which are heads is 4/10 = 0.4. For any one flip of the coin, I cannot be sure whether it will show up heads or tails. However, the long-term proportion should be ½.

The Law of Large Numbers

The Law of Large Numbers As the number of repetitions of a random experiment increases. . .

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

In probability, an experiment is any process that can be repeated in which the results are uncertain. A simple event is any single outcome from a probability experiment. Each simple event is denoted ei.

To make the ideas in the examples more precise, we need some definitions.

To make the ideas in the examples more precise, we need some definitions. The definitions apply to the simplest situations where probability applies.

To make the ideas in the examples more precise, we need some definitions. The definitions apply to the simplest situations where probability applies. E.g., flipping coins, card games, roulette, etc.

To make the ideas in the examples more precise, we need some definitions. The definitions apply to the simplest situations where probability applies. E.g., flipping coins, card games, roulette, etc. We will look at these kinds of situations first.

DEFINITIONS:

DEFINITIONS: A random experiment is any process that can be repeated in which the results are uncertain.

DEFINITIONS: A random experiment is any process that can be repeated in which the results are uncertain. A simple event is any single outcome from a probability experiment.

DEFINITIONS: A random experiment is any process that can be repeated in which the results are uncertain. A simple event is any single outcome from a probability experiment. Each simple event is denoted ei.

Sample space, S, of an experiment:

Sample space, S, of an experiment: The collection of all possible simple events.

Sample space, S, of an experiment: The collection of all possible simple events. In other words, the sample space is a list of all possible outcomes of a probability experiment.

An event is any collection of outcomes from a probability experiment.

An event is any collection of outcomes from a probability experiment. An event consists of one or more simple events.

An event is any collection of outcomes from a probability experiment. An event consists of one or more simple events. Events are denoted using capital letters such as E.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin.

Consider the experiment of flipping a coin. The sample space S is: EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is:

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails}

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} The simple events are:

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} The simple events are: (1) Coin lands Heads

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} The simple events are: (1) Coin lands Heads (2) Coin lands Tails

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} The simple events are: (1) Coin lands Heads (2) Coin lands Tails An event E might be:

Consider the experiment of flipping a coin. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} The simple events are: (1) Coin lands Heads (2) Coin lands Tails An event E might be: Coin lands Heads or Tails

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are:

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3,

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6,

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6, An event E:

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6, An event E: roll an even number

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6, An event E: roll an even number {2,4,6}.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6, An event E: roll an even number {2,4,6}. Another event E:

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6, An event E: roll an even number {2,4,6}. Another event E: roll a number greater than 4

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. The simple events are: roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6, An event E: roll an even number {2,4,6}. Another event E: roll a number greater than 4 {5,6}.

The probability of an event E, is the likelihood of that event occurring.

The probability of an event E, is the likelihood of that event occurring. The probability of E is denoted: P(E)

Properties of Probabilities The probability of any event E, P(E), must be between 0 and 1 inclusive.

Properties of Probabilities The probability of any event E, P(E), must be between 0 and 1 inclusive. 0 < P(E) < 1.

Properties of Probabilities The probability of any event E, P(E), must be between 0 and 1 inclusive. 0 < P(E) < 1. 2. If an event is impossible, the probability of the event is 0.

Properties of Probabilities The probability of any event E, P(E), must be between 0 and 1 inclusive. 0 < P(E) < 1. 2. If an event is impossible, the probability of the event is 0. 3. If an event is a certainty, the probability of the event is 1.

Properties of Probabilities If S = {e1, e2, …, en},

Properties of Probabilities If S = {e1, e2, …, en}, P(e1) + P(e2) + … + P(en) = 1.

Properties of Probabilities If S = {e1, e2, …, en}, P(e1) + P(e2) + … + P(en) = 1. This last property is just another way of saying the probability that something happens is a certainty.

An unusual event is an event that has a probability close to zero.

An unusual event is an event that has a probability close to zero. An impossible event has probability equal to zero.

An unusual event is an event that has a probability close to zero. An impossible event has probability equal to zero. A likely event has probability close to one.

An unusual event is an event that has a probability close to zero. An impossible event has probability equal to zero. A likely event has probability close to one. A certain event has probability equal to one.

The book mentions three methods for determining the probability of an event:

The book mentions three methods for determining the probability of an event: the classical method

The book mentions three methods for determining the probability of an event: the classical method Will cover this in more detail – Pascal and Fermat

The book mentions three methods for determining the probability of an event: the classical method Will cover this in more detail – Pascal and Fermat (2) the empirical method

The book mentions three methods for determining the probability of an event: the classical method Will cover this in more detail – Pascal and Fermat (2) the empirical method Essentially consists of running a random experiment several times and measuring the proportion of times each outcome occurs.

The book mentions three methods for determining the probability of an event: the classical method Will cover this in more detail – Pascal and Fermat (2) the empirical method Essentially consists of running a random experiment several times and measuring the proportion of times each outcome occurs. (3) the subjective method

The book mentions three methods for determining the probability of an event: the classical method Will cover this in more detail – Pascal and Fermat (2) the empirical method Essentially consists of running a random experiment several times and measuring the proportion of times each outcome occurs. (3) the subjective method Judge based on level of confidence that something will occur – e.g., what sports odds-makers do.

The book mentions three methods for determining the probability of an event: the classical method Will cover this in more detail – Pascal and Fermat (2) the empirical method Essentially consists of running a random experiment several times and measuring the proportion of times each outcome occurs. (3) the subjective method Judge based on level of confidence that something will occur – e.g., what sports odds-makers do. This is what humans do instinctually.

The classical method of computing probabilities requires equally likely outcomes.

The classical method of computing probabilities requires equally likely outcomes. An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring.

Computing Probability Using the Classical Method

Computing Probability Using the Classical Method An experiment has n equally likely simple events.

Computing Probability Using the Classical Method An experiment has n equally likely simple events. The number of ways that an event E can occur is m.

Computing Probability Using the Classical Method An experiment has n equally likely simple events. The number of ways that an event E can occur is m. Then the probability of E, P(E), is:

Computing Probability Using the Classical Method An experiment has n equally likely simple events. The number of ways that an event E can occur is m. Then the probability of E, P(E), is:

Computing Probability Using the Classical Method If S is the sample space of the experiment.

Computing Probability Using the Classical Method If S is the sample space of the experiment. N(S) denotes the number of simple events in S.

Computing Probability Using the Classical Method If S is the sample space of the experiment. N(S) denotes the number of simple events in S. N(E) denotes the number of simple events in E.

Computing Probability Using the Classical Method If S is the sample space of the experiment. N(S) denotes the number of simple events in S. N(E) denotes the number of simple events in E.

EXAMPLE: Classical Probability

EXAMPLE: Classical Probability Consider the experiment of flipping a coin.

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails}

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.)

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) =

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) = ½

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) = ½ P(T) = ½

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) = ½ P(T) = ½ Let E be the event: H or T {H,T}.

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) = ½ P(T) = ½ Let E be the event: H or T {H,T}. P(E) = 1.

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) = ½ P(T) = ½ Let E be the event: H or T {H,T}. P(E) = 1. Let E be the event that we get neither H nor T.

EXAMPLE: Classical Probability Consider the experiment of flipping a coin. The sample space S is: {Heads, Tails} Each simple event is equally likely (fair coin.) P(H) = ½ P(T) = ½ Let E be the event: H or T {H,T}. P(E) = 1. Let E be the event that we get neither H nor T. P(E) = 0.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}. P(E) = N(E)/N(S)

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}. P(E) = N(E)/N(S) = 3/6 = 1/2

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}. P(E) = N(E)/N(S) = 3/6 = 1/2 Let E be the event roll greater than 4.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}. P(E) = N(E)/N(S) = 3/6 = 1/2 Let E be the event roll greater than 4. E = {5,6}.

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}. P(E) = N(E)/N(S) = 3/6 = 1/2 Let E be the event roll greater than 4. E = {5,6}. P(E) = N(E)/N(S)

EXAMPLE. Identifying Events and the Sample EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the experiment of rolling a die. The sample space S = {1, 2, 3, 4, 5, 6}. N(S) = 6. P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Let E be the event roll is even. E = {2,4,6}. P(E) = N(E)/N(S) = 3/6 = 1/2 Let E be the event roll greater than 4. E = {5,6}. P(E) = N(E)/N(S) = 2/6 = 1/3

A probability distribution is just the list of probabilities for each simple event in the sample space.

A probability distribution is just the list of probabilities for each simple event in the sample space. For instance, the probability distribution of the fair coin is:

A probability distribution is just the list of probabilities for each simple event in the sample space. For instance, the probability distribution of the fair coin is: P(H) = ½

For instance, the probability distribution of the fair coin is: A probability distribution is just the list of probabilities for each simple event in the sample space. For instance, the probability distribution of the fair coin is: P(H) = ½ P(T) = ½

For instance, the probability distribution of the fair coin is: A probability distribution is just the list of probabilities for each simple event in the sample space. For instance, the probability distribution of the fair coin is: P(H) = ½ P(T) = ½ The probability distribution of the fair die is:

For instance, the probability distribution of the fair coin is: A probability distribution is just the list of probabilities for each simple event in the sample space. For instance, the probability distribution of the fair coin is: P(H) = ½ P(T) = ½ The probability distribution of the fair die is: P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

The classical probability distribution, where each simple event is equally likely to occur, is often called the uniform distribution.

The classical probability distribution, where each simple event is equally likely to occur, is often called the uniform distribution. The assumption that each simple event is equally likely is often not valid in real-life situations.

The classical probability distribution, where each simple event is equally likely to occur, is often called the uniform distribution. The assumption that each simple event is equally likely is often not valid in real-life situations. It may be possible to obtain another distribution from theoretical considerations (binomial, etc.)

The classical probability distribution, where each simple event is equally likely to occur, is often called the uniform distribution. The assumption that each simple event is equally likely is often not valid in real-life situations. It may be possible to obtain another distribution from theoretical considerations (binomial, etc.) However, if this is not the case we can obtain an approximation to the probability distribution empirically.

EXAMPLE Using Relative Frequencies to Approximate Probabilities The following data represent the number of homes with various types of home heating fuels based on a survey of 1,000 homes.

What is the approximate probability of each heating source?

What is the approximate probability of each heating source? 504/1000 = .504

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307 94/1000 = .094

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307 94/1000 = .094 2/1000 = .002

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307 94/1000 = .094 2/1000 = .002 17/1000 = .017

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307 94/1000 = .094 2/1000 = .002 17/1000 = .017 1/1000 = .001

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307 94/1000 = .094 2/1000 = .002 17/1000 = .017 1/1000 = .001 4/1000 = .004

What is the approximate probability of each heating source? 504/1000 = .504 64/1000 = .064 307/1000 = .307 94/1000 = .094 2/1000 = .002 17/1000 = .017 1/1000 = .001 4/1000 = .004 7/1000 = .007