Chapter 3.1 Probability Students will learn several ways to model situations involving probability, such as tree diagrams and area models. They will formalize methids for computing probabilities of unions, intersections and complements of events. Then they will calculate expected value in games of chance.

3.1.1 Using an Area Model

Likelihood something will happen Probability Likelihood something will happen Can’t be negative because we can’t have a negative number or outcomes or want a negative amount of something to happen

Terms we need to know for Probability Event: any outcome or set of outcomes from a situation Successful Event: set of all outcomes that are of interest in a given situation Rolling a 5 is an event If you win when you roll an even number then {2,4,6} would be successful events Sample Space: all possible outcomes Roll a dies {1,2,3,4,5,6}

More terms Probability: The likelihood that an event will occur Written as a ratio, decimal or percent An event that is certain to happen has a probability of 1 An event that has no chance of happening has a probability of 0 Theoretical Probability: probability that is mathematically calculated When each of the outcomes in the same space has an equally likely chance of occurring

Theoretical Probability When the outcomes in the sample space of an experiment have the same chance of occurring, the outcomes are equally likely Number of ways your event can occur total number of outcomes Using the math to find the probability

Example: this is in your notes A bag of marbles contains 8 yellow, 2 red, and 10 green marbles. An experiment consists of selecting one marble at random from the bag. Find the theoretical probability of each outcome. Selecting a yellow marble Selecting a red or a yellow marble

Experimental Probability An experiment is an activity involving chance Use the data from doing the experiment to determine the probability Number of times event occurs number of trials

Example: this is in your notes An experiment consists of randomly selecting marbles from a bag. Use the results in the table to find the experimental probability of each event. Select a green marble Not selecting a white

Area Model Similar to the idea of a 2 way table Top is one of the outcomes with probability for it to happen or not Side is the other outcome with probability for it to happen or not The Area model then represents all the combinations and probabilities for each to happen

Area Model examples Pg 145 read through 3-1 as a group and work through a, b and c When done and I have checked this go onto How do you fill in the table Pg 146 read and work through 3-2 as a group, work through a,b,c Set up a new area model How do you again fill in the table This process reminds me of the lattice work a lot of you do for multiplication and for multiplying polynomials

More examples How would you set up the table for 3-3 What about 3-4 Try 3-6 Roll and Win

3.1.2 Using a Tree Diagram

Tree Diagram Using the roots of a tree to give you all the paths and possible outcomes in a given situation Multiply along the path (similar to finding the boxes) Add all paths together and you should get 1 I like this it seems more visual to me

Tree Diagram Each branch represents an outcome or event Write the probability for that event Next branch is different outcome or event Write the probability Then multiply across

More Terms Independent Event: does one event affect the other Flipping a coin and rolling a die Dependent Event: does one event affect the other Picking something out of a back and not putting it back

Independent Probability If the occurrence of one event does not affect the probability of the other If A and B are independent events then P(A and B)=P(A) * P(B) Rolling a die and flipping a coin Pick a card replacing it and then picking another card If two or more events are independent then multiply their individual probabilities

Dependent Probability If the occurrence of one event does affect the probability of the other If A and B are dependent events then P(A and B) = P(A)*P(B after A) In other words the second probability changes as a result of the first Outcomes goes down by 1 and numerator could change depending on the event You have a situation where you do not replace something Pick a marble, not replacing it and then picking another marble

Determine if Independent or Dependent A die is rolled two times. The numbers 1-20 are written on pieces of paper and put in a box. Two pieces are randomly selected and not replaced. Selecting a marble and replacing it Selecting a marble and not replacing it Selecting two boys from this class to do an errand

3-12 Looking at 3-12 on page 150 Could you make an area model for this Otherwise look at it like a tree First set of branches is spinner 1 Second set of branches is spinner 2 Write the probability along each branch