Probability

Definitions Probability: The chance of an event occurring. Probability Experiments: A process that leads to well- defined results called outcomes. Examples of Experiments: Flipping a coin, Rolling a die, Picking a card from a deck of cards. Outcomes: The result of a single trial of a probability experiment. Sample Space: The set of the outcomes of a probability experiment.

Definitions Events: Consist of one or more outcomes of a probability experiment. 2 types of events Simple Event: An event with one outcome. (Rolling a 3 on a die) Compound Event: An event with more than one outcome. ( Rolling an even number on a die)

Classical Probability Classical Probability: uses sample spaces to determine the numerical probability that an event will happen. Assumes that all outcomes in the sample space are equally likely to occur. number of outcomes in E total number of outcomes in sample space Denoted by P(E).

Classical Examples What is the probability of rolling an even number on a die? What is the probability of picking a king from a deck of cards? What is the probability of rolling a number less than 7 on a single die? All probabilities must fall within the following range. ( 0 ≤ n ≤ 1 )

Complementary Events Complementary Events: The set of outcomes in the sample space that are not included in the outcomes of event E. Denoted by Ē (read “ E Bar” ) Example 1: If we roll an even number on a die its complement is rolling an odd. Example 2: Picking a red card from a deck its complement is picking a black card. Rule: P (Ē) = 1 – P(E)

Empirical Probability Empirical Probability: Relies on actual experiments to determine the likelihood of an event. Using data from actual survey or other type of research. Based on observation. Frequency for the class total frequencies in the distribution P (E) = f n

Empirical Examples 20 fans from a Rutgers Football game were asked how many home games they had been to this season. The results are shown below. Games attendedFrequency

Law of Large Numbers Law of Large Numbers: The more times that a experiment is conducted the more likely the empirical probability will approach the theoretical probability. Example: Tossing a coin 50 or more times should lead to the theoretical probability of getting a tail = ½.

Subjective Probability Subjective probability: uses a probability value based on an educated guess or estimate, employing opinions and inexact information. Example 1: A doctor diagnosing that a patient has a 30% chance of getting the flu this winter. Example 2: A weather forecaster stating that there is a 60% chance of rain tomorrow.

Addition Rule for Probability Addition Rule: Determines the probability of a compound event. An event that has 2 or more outcomes. Example 1: Selecting a 6 or an 8 from a deck of cards. Example 2: Getting a 1 or a 5 when rolling a die. Mutually exclusive events: Two events that can’t occur at the same time. Not mutually exclusive: Two events that can occur at the same time.

Exclusive or Not Determine if the following events are mutually exclusive or not if a single card is chosen from a deck. 1.Getting a 7 or getting a 9. 2.Getting a heart or a Getting a face card or a black card. 4.Getting a 2 or getting a face card.

Addition Rule 1 When 2 events are mutually exclusive the probability that A or B will occur is P(A or B) = P(A) + P(B) Example 1: Probability of rolling a 3 or a 4 on single die. Example 2: Probability of picking a queen or the Ace of Spades when pulling a card from a deck Example 3: 7 cashiers, 2 managers, and 2 stock boys work at a local store. What is the probability if a person is selected at random that they are a cashier or a stock boy?

Addition Rule 2 When 2 events are not mutually exclusive the probability that A or B will occur is: P(A or B) = P(A) + P(B) – P(A and B) Example 1: Probability of selecting a 6 or a black card from a deck. Example 2: Probability of rolling a 2 or an even number on a single die. Example 3: In a class there are 16 seniors and 11 juniors. 7 of the seniors are male and 6 or the juniors are female. If a student is selected at random what is the probability of selecting a senior or a male? Example 4: The probability of a tourist visiting the Empire State Building is 0.64 and of visiting Grand Central Station is The probability of visiting both on the same day is Find the probability that a tourist visits the Empire State Building or visits Grand Central Station.

Multiplication Rule 1 Used to find the probability of 2 or more events that occur in sequence. Example: Rolling a die then flipping a coin one can find the probability of rolling a 3 and getting a tail. Independent Events: Events that do not affect each other. Their probabilities of occurring are also not affected. P(A and B) = P(A) ● P(B)

Rule 1 Examples Example 1: A spinner with 5 sections, numbered 1 -5 is spun and a card is picked from a deck. What is the probability of the spinner landing on 3 and picking a heart? Example 2: A coin is tossed and a card from a deck is picked. What is the probability of landing on heads and getting a face card? Example 3: Two dice are rolled, one at a time, what’s the probability of rolling a 5 and another 5?

Multiplication Rule 2 Used to find the probability of 2 or more events that occur in sequence. Example: Probability of picking a king from a deck then picking a 6 if not replacing the first card. Dependent Events: When the outcomes or occurrence of the 1st event affects the 2 nd event in such a way that the probability is changed. Notation: P( B│A) This does not mean B divided by A. It means that the probability event B occurs given that event A has already happened. In example above P(B│A) = 4/51 Formula: P( A and B) = P (A) ● P(B│A)

Rule 2 Examples Example 1: What is the probability of picking an ace and ace if the first card is not replaced? Example 2: 3 blue, 6 white, 2 red marbles all in a bag. What is the probability of picking a blue, white, and red marble if marbles are not replaced? Example 3: In a recent survey 49% of people said they had renter’s insurance with Geico. Of those 22% said they also had car insurance with them. If a person is selected at random, what’s the probability that they have both?

Conditional Probability Conditional Probability: Imposes a condition on the problem. The word “given” imposes the condition. The probability that the 2 nd event B occurs given the 1 st event A has occurred can be found by dividing the probability that both events occurred by the probability that the 1 st event has occurred. Formula: P(B│A) = P (A and B) ÷ P (A)

Conditional Examples Example 1: At a country club 52% of the members play bridge and swim, and 65% swim. If a member is selected at random, find the probability that the member plays bridge, given that the member swims. Example 2: At a small college the probability a students takes math and physics is The probability that a student takes math is Find the probability that the student is taking physics given that he or she is taking math.

Conditional Examples Use the chart to answer the questions. If a person is selected at random find the probability the person owns a credit card, given that the person is employed. Find the probability the person is unemployed given that the person does not own a credit card. Employment StatusOwns a CardDoesn't Own a Card Employed1023 Unemployed1535

Tree Diagram Problems Example: A electronics company has 2 factories that produce televisions. Factory A produces 60% and Factory B produces 40%. 8% of the televisions from Factory A are defective while 15% from Factory B are defective. If a tv is selected at random, find the probability it is defective.

“At Least” Probabilities Can be tedious to find the result of each possibility and then add them together. Easier to find the complement of the event and then subtract that from 1. Example 1: Find the probability of getting at least one King if you draw 4 cards from a deck. If the card is replaced. Then if not replaced. Example 2: A coin is tossed 3 times. What is the probability of at least one head?