The Sample Space with Tables and Tree Diagrams

all of the possibilities Remember that the sample space shows all of the possibilities or outcomes of an event!

Charts and Tree Diagrams Most common ways to display a sample space is either through a CHART or a TREE DIAGRAM. Chart – organized table of possible outcomes. Tree Diagram – uses line segments coming from starting point to each option of outcome.

Example (sample spaces chart) Find the sample space for rolling two dice. Die1 Die 2 1 2 3 4 5 6 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Tree Diagram Use a tree diagram to find the sample space for the gender of three children in a family.

For example – a fair coin is spun twice 1st 2nd H HH Possible Outcomes H T HT H TH T T TT

So the possibilities of a meal are Choose a meal Pudding Ice Cream Apple Pie Main course Salad Egg & Chips Pizza IC AP So the possibilities of a meal are S, IC S, AP E, IC E, AP P, IC P, AP S E P IC AP IC AP

For example – 10 coloured beads in a bag – 3 Red, 2 Blue, 5 Green For example – 10 coloured beads in a bag – 3 Red, 2 Blue, 5 Green. One taken, colour noted, returned to bag, then a second taken. 1st 2nd R RR B RB R G RG R BR INDEPENDENT EVENTS B B BB G BG R GR G GB B G GG

Let’s learn about probability and chance!

What is probability? Probability is the measure of how likely an event or outcome is. Different events have different probabilities!

How do we describe probability? You can describe the probability of an event with the following terms: certain (the event is definitely going to happen) likely (the event will probably happen, but not definitely) unlikely (the event will probably not happen, but it might) impossible (the event is definitely not going to happen) Can you think of examples of each type of event?

How do we express probabilities? Usually, we express probabilities as fractions. The numerator is the number of ways the event can occur. The denominator is the number of possible events that could occur. Let’s look at an example!

What is the probability the spinner will land on the number 3? 1 2 4 3

Ask yourself the following questions: 1 Ask yourself the following questions: 1. How many 3’s are on the spinner? 2. How many possible numbers could the spinner land on? 1 1 2 4 4 3

What is the probability the die will land on an even number? Remember, a die has six sides. Numbers 1, 2, 3, 4, 5, and 6 are each depicted once on the die.

Ask yourself the following questions: 1 Ask yourself the following questions: 1. How many even numbers are on the die? 2. How many possible numbers could the die land on? 3 6

What is the probability that I will choose a red marble? In this bag of marbles, there are: 3 red marbles 2 white marbles 1 purple marble 4 green marbles

Ask yourself the following questions: 1 Ask yourself the following questions: 1. How many red marbles are in the bag? 2. How many marbles are in the bag IN ALL? 3 10

Probability Is Fun! Next time you’re playing a board game or a carnival game, think about the probability of the situation! In the next few days, you will be conducting probability experiments and designing probability games! Get ready to use your probability power! 

Probability Complement

Definition Complementary Events (1/3) The Complement is the event that describes all other possible events. Example: E = {winning a race} Complement of E = {NOT winning a race}

Complementary Events (2/3) The Sum of the Probabilities of an Event and its Complement add to ‘1’. Example: For a dice 2 6 1 3 P(>4) = = 4 6 2 3 P( Complement >4) = P(≤4) = = 1 3 2 3 P(>4) + P( Complement >4) = + = 1

Complementary Events (3/3) To find the Probability of an Event, subtract the probability of its Complement from ‘1’. Example: For a standard pack of cards P(Spot Card) = 1 - P( Not Spot Card) 12 52 = 1 - 10 13 =

Compound Event Probability

Probability of Compound Events Review: A Simple Event has only one outcome. New Info: A Compound Event is a combination of at least two simple events. There are two kinds of compound events: Independent Events - When the outcome of the first event has no influence on the likelihood of a future event occurring. Dependent Events - When the outcome of the first event reduces the amount of possible outcomes (and as a result, the likelihood) of future event(s). Vocabulary: Simply present the information to be copied in notes. How the math is done will be covered in other slides. Definitions of key words should be given: Compound, Independent, Dependent.

Review of a Simple Event What is the probability of rolling less than a 5? Review: of what they previously covered. Note: Skip this slide if the review is not necessary

multiply the probabilities of the individual events. Compound Events To determine the probability of a compound event, multiply the probabilities of the individual events. Example: What is the probability of drawing a Queen and then a King? There are 52 cards in a deck and in it are 4 Queens as well as 4 Kings. So the odds are 4/52 (simplified to 1/13) Present Concept 1 of 5 Achieves Learning Objectives #1 & 2: Students will be able to adapt a trial to include a second trial (Compound Events) Students will be able to determine probability of a Compound Event by combining the theoretical probabilities using multiplication. Answer to Guiding Question #2: How do you mathematically determine probability when a second trial is added? You multiply the Probabilities of both events.  

Key Difference! 1/13 = 0.063 (about six hundredths) 1/169 = 0.0059 Probability of a single event Probability of the compound event This one is less than 1/10… 1/13 = 0.063 (about six hundredths) 1/169 = 0.0059 (about six thousandths) of this one! Present Concept 2 of 5 Answer to Guiding Question #1 What impact does a second trial have on the probability of success? It reduces it greatly. In this example, it is less than one tenth the chance!  

Spinner Probability A fair spinner is one on which all the sections are the same size. A fair competition is one in which everyone has the same probability of winning. To win, you must first spin a number higher than 7, and then then spin an even number. Your probability … Present Concept 3 of 5 Consider if your students know how to use a spinner like the one pictured. Explain the blue text if necessary. Next, fill in the blanks to demonstrate how it done. Ask for students to come up with what numbers go in places you point to. As students give suggestions and the correct number is given, ask a follow-up question such as, “Why does ‘John’ say ‘10’ goes here?” Note: If you wish to fill in the answers yourself and, perhaps, cross cancel, delete the unneeded features on the slide. Students will demonstrate their understanding of GUIDING QUESTION #2: How do you mathematically determine probability when a second trial is added?

Dependent Events Abbreviations are used to illustrate what happens when we solve probabilities. P: Means total probability A: Refers to the first event B: Refers to the second event Probability that 2 independent events will take place: P(A&B) = P(A)P(B) The probability of A and B both happening equals the probability of A times the probability of B. Probability that 2 dependent events will take place: P(A&B) + P(A)P (B following A) The probability of A & B happening equals the probability of A times the probability of B after A has happened. Present Concept 4 of 5 As you work through the the algorithms, be certain to check for understanding A better opportunity to demonstrate the Dependent Event is part of the next slide Achieves Learning Objectives #3: Students will understand that some compound trials are Independent and some are Dependent.

Note the pattern: P(A&B) = P(A)P(B following A) Dependent Events “Example: You share your last jelly beans with a friend. There are 12 jelly beans left over. Both of you like the red beans most. There are only 2. Your friend is next get to take two beans. What is the probability that she can get both red beans? Present Concept 5 of 5 Answer to Guiding Question #3: What is the mathematical influence when the outcome of the first trial changes the possible outcomes of the second trial? The “Possible Outcomes” of the second trial is reduced by the amount of items removed as a result of the first trial. Note: Photo of jelly beans was copied from: http://pixabay.com/p-321186/?no_redirect. A note on the site indicated that the photo could be freely used without any notations as to its source. Note the pattern: P(A&B) = P(A)P(B following A)

Practice There are 12 marbles in the bag: 2 Blue 3 Yellow 3 Green 4 Red Your friend is to reach in and try to take out a yellow marble. He will keep it out. You will try to the same with a red marble. What is the probability of success? Guided Practice Students will demonstrate their understanding of Guiding Question #3: What is the mathematical influence when the outcome of the first trial changes the possible outcomes of the second trial? Kagan Engagement Idea: Simultaneous Round Table: In teams, students each write a response on their own piece of paper. Students then pass their papers clockwise so each teammate can add to the prior responses. (In this situation, have students write only the next portion of the problem on the paper before passing it along, i.e.: first “1/2,” then “x,” then “2/3,” and so on.) Circulate around the room providing guidance. After students have finished, return to board and work out the problem on the board. Again, If you wish to fill in the answers yourself and, perhaps, cross cancel, delete the unneeded features on the slide.