A casino claims that its roulette wheel is truly random A casino claims that its roulette wheel is truly random. What should that mean? Black has come up as the winner 4 times in a row, are chances better for betting on black or red the next time, why? There are 37 spaces on the wheel, what is the probability that the ball will land on Zero? A number divisible by three? A black even number? Roulette Warm-up

WARM- up Clear desks for Ch 13 Quiz. After we will be taking notes on Ch 14 so have notebook paper ready. Don’t forget – Helicopter Projects are due next class, whether you are here or absent!! WARM- up

Today we will review the vocabulary of Ch 14, then learn how to calculate compound probability of disjoint and independent events. Objective

Law of Large Numbers (LLN) the long-run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases. For example, consider flipping a fair coin. Law of Large Numbers (LLN)

Calculate the probability of six compound probabilities; three disjoint and three independent. Closing Product

I flip a fair coin one time I flip a fair coin one time. What is the likelihood that it will be heads? I flip a fair coin 10 times. It is tails every time. On the 11th flip what is the likelihood that it will come up tails again? If I flip a coin 1000 times how many times will it come up heads and how many times will it come up tails? LLN

Commercial airplanes have an excellent safety record, however, crashes still occur. In the weeks following a fatal crash, people are afraid of flying. A Travel agent tries to calm a traveler’s fear by saying it’s the safest time to fly because it is unlikely two crashes will occur so closely together. How about the converse? Example

The proportion of times an outcome occurs in a very long series of repetitions. P(A) = # of times event A occurs total number of occurrences Probability

Rules of Probability Two requirements for a probability: A probability is a number between 0 and 1. Therefore, For any event A, 0 ≤ P(A) ≤ 1. Rules of Probability

Rules of Probability “Something has to happen rule”: The probability of the set of all possible outcomes of a trial must be 1. P(S) = 1 (S represents the set of all possible outcomes.) Rules of Probability

SPINNER EXAMPLE Red Yellow Green Blue 1 0.25 2 0.5 0.3 0.2 0.10 3 .20 .4 .5 .3 4 5 .1 .2 1.2 -1.5 SPINNER EXAMPLE

Rules of Probability Complement Rule: The set of outcomes that are not in the event A is called the complement of A, denoted AC. The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC) Rules of Probability

EXAMPLE A bag of marbles has 3 yellow, 4 blue and 5 green. What is the probability of not blue? What is the probability of not yellow or green? What is another way to write this? EXAMPLE

Events in which the outcome of one trial do not influence the outcome of the other. Ex. Drawing a card from a deck, flipping a coin, etc. Independent Events

The probability that two or more independent events will occur. Used for “and” problems Read, “the union of A and B” Multiplication Rule

For example, when tossing a fair coin twice, the probability of getting a 'Head' on the first and then getting a 'Tail' on the second is P(H and T) = P(H) × P(T) P(H and T) = 0.5 × 0.5 P(H and T) = 0.25 Example

A bag of marbles has 3 yellow, 4 blue and 5 green A bag of marbles has 3 yellow, 4 blue and 5 green. Every time you draw a marble you must replace it before drawing another one. What is the probability of drawing a yellow then a green? What is the probability 3 blues in a row? What is the probability of drawing a yellow, a blue and a green? EXAMPLE

Disjoint (mutually exclusive) Events that cannot occur at the same time. Flipping a single coin and getting a head and a tail. Drawing a 9 and even card from a deck of cards. Disjoint (mutually exclusive)

For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events. P(A or B) = P(A) + P(B), provided that A and B are disjoint. Addition Rule

A bag of marbles has 3 yellow, 4 blue and 5 green A bag of marbles has 3 yellow, 4 blue and 5 green. Every time you draw a marble you must replace it before drawing another one. What is the probability of drawing a yellow or a green? What is the probability blue or a green? EXAMPLE

P(at least A) = 1- P(no A) At least problems

Example If you pick three in a row, what is the probability that… M& M’s reportedly make 20% of the candies yellow, 20 % red, 10% each of orange, blue and green and the rest are brown. If you pick three in a row, what is the probability that… They are all brown The third one is the first one that is red? None are yellow? At least one is green? Example