Chapter 6: Probability and the Study of Randomness

Chapter 6: Probabilityand the Study of Randomness Mmm, peanut butter!! 6.2 Probability Models

The Language of Probability pg 408 Random Event • A random event has outcomes that we can’t predict, but nonetheless has a regular distribution in many repetitions Probability • The probability of an event is the proportion of times the event occurs in many repeated trials of random event

Probability = long run relative frequency What does it mean when we say “the probability of heads in a fair flip of a coin is 0.5”? Probability Applet Over time, the relative frequency of a random event will “settle down” to the probability of the event.

Developing a Probability Model To get mathematical about randomness develop a model. • Make a list of all possible outcomes • Obtain a probability for each outcome

A List all possible outcomes The Sample Space Flip two coins S= {HH, HT, TH, TT} Family of 3 children S={BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}

2 ways to make sure you count all possible outcomes • Tree Diagram: What are the possible outcomes for the birth order of a family of 3 children? • B G • B G B G • B G B G B G B G

Multiplication Rule: If you can do one task in x number of ways and a second task in y number of ways then both tasks can be done in x·y number of ways. How many outcomes are possible if you roll a die and flip 2 coins? HH1, TT3, HT6, TH4, etc 2 x 2 x 6 = 24 different ways

HW #72: Read pg 417 – 431 6.23, 6.24, 6.28, 6.29, 6.33, 6.36

Any probability is a number between 0 and 1. An event with 0 probability never occurs. An event with probability 1 occurs all the time Probability P(A) of any event A 0 ≤ P(A) ≤ 1 Get to know the symbols 5 Basic Probability Rules

Probability Rules • All possible outcomes together must have probability = 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must equal 1. This is the sample space S. • If S is the sample space then P(S) = 1

Disjoint = Mutually Exclusive Disjoint (or mutually exclusive) events can not happen at the same time Which of the following events are not disjoint? • Wearing tennis shoes, wearing flip flops • Wearing tennis shoes, wearing glasses • Graduating in 2008, Graduating in 2009

S A B S A B Probability Rules 3. P(A or B) = P(A) +P(B) P(A U B) = P(A) + P(B) if events are disjoint Events A and B are disjoint P(A or B) = P(A) + P(B) Events A and B are not disjoint P(A or B)  P(A) + P(B)

Ac A Probability Rules 4. The probability that an event does not occur is 1 minus the probability that the event does occur. This event is called the complement. • P(Ac) = 1 – P(A) Venn Diagram

Independent Events • Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. Which events are independent? • wearing a ski hat, wearing gloves • enrolled in Spanish class, enrolled in Health • graduating from high school, earning minimum wage

S A B Probability Rules 5. P(A and B) = P(A)*P(B) if the events are independent Multiplication Rule: P(A ∩ B)

Chapter 6: Probability and the Study of Randomness