Unit 1 Review MDM 4U Chapters 4.1 – 4.5

Test Format 20 MC K/U (answer 4 of 4) APP (answer 4 of 6) TIPS (answer 2 of 3) COMM (15% - using formulas, good form, rounding, concluding statements)

Steps to Solving Probability Problems G = Given: List knowns, draw diagram / table / list R = Required to find: Identify the unknown, look for keywords A = Analyze: Write the formula or explain your thinking S = Solve: Plug knowns into formula and/or compute P = Present: State your answer using a sentence

U1 Learning Goals Calculate experimental probability given the results of an experiment Design a simulation for a given scenario Calculate the theoretical probability for simple or compound events Recognize and calculate conditional probability Weather, pro athletes, timed traffic signals, dealing cards Recognize and calculate intersection based on a conditional probability (dealing cards) Use the multiplicative principle for union (or intersection) Create or use a Venn diagram to calculate probability Construct a tree diagram or outcome table and use it to calculate probability

4.1 Intro to Simulations and Experimental Probability Design a simulation to model the probability of an event Ex: design a simulation to determine the experimental probability that more than one of 5 keyboards chosen in a class will be defective if we know that 25% are defective 1. Get a well-shuffled deck of cards, choosing clubs to represent a defective keyboard 2. Choose 5 cards with replacement and record how many are clubs 3. Repeat a large number of times and calculate the experimental probability

4.2 Theoretical Probability Calculate the probability of an event or its complement Ex: What is the probability of randomly choosing a male from a class of 30 students if 10 are female? P(A) = n(A) = 20 = 0.67 n(S) 30

4.2 Theoretical Probability Ex: Calculate the probability of not throwing a total of four with 3 dice There are 63 = 216 possible outcomes with three dice Only 3 outcomes produce a 4 P(sum = 4) = 3_ 216 probability of not throwing a sum of 4 is: 1 – 3 = 213 = 0.986 216 216

4.3 Finding Probability Using Sets Recognize the different types of sets (union, intersection, complement) Utilize: The Additive Principle for the Union of Two Sets: n(A U B) = n(A) + n(B) – n(A ∩ B) P(A U B) = P(A) + P(B) – P(A ∩ B) The Additive Principle for the Intersection of Two Sets: n(A ∩ B) = n(A) + n(B) – n(A U B) P(A ∩ B) = P(A) + P(B) – P(A U B)

4.3 Finding Probability Using Sets Work with Venn diagrams ex: Create a Venn diagram illustrating the sets of face cards and red cards red & face = 6 face = 6 red = 20 S = 52

4.3 Finding Probability Using Sets Ex: What is the probability of drawing a red card or a face card Ans: P(A U B) = P(A) + P(B) – P(A ∩ B) P(red or face) = P(red) + P(face) – P(red and face) = 26/52 + 12/52 – 6/52 = 32/52 = 0.615

4.3 Finding Probability Using Sets What is n(B υ C) 2+8+3+3+6+2+1+8+1 = 34 What is P(A∩B∩C)? n(A∩B∩C) = 3 = 0.07 n(S) 43

4.4 Conditional Probability 100 Students surveyed Course Taken No. of students English 80 Mathematics 33 French 68 English and Mathematics 30 French and Mathematics 6 English and French 50 All three courses 5 What is the probability that a student takes Mathematics given that he or she also takes English?

4.4 Conditional Probability E 4.4 Conditional Probability 17 45 1 5 2 5 25

4.4 Conditional Probability To answer the question in (b), we need to find P(Math|English). We know... P(Math|English) = P(Math ∩ English) P(English) Therefore… P(Math|English) = 30 / 100 = 30 x 100 = 3 80 / 100 100 80 8

4.4 Conditional Probability Calculate the probability of event B occurring, given that A has occurred Need P(B|A) and P(A) Use the multiplicative law for conditional probability Ex: What is the probability of drawing a jack and a queen in sequence, given no replacement? P(1J ∩ 2Q) = P(2Q | 1J) x P(1J) = 4 x 4 = 16 = 0.006 51 52 2 652

4.5 Tree Diagrams and Outcome Tables A sock drawer has a red, a green and a blue sock You pull out one sock, replace it and pull another out Draw a tree diagram representing the possible outcomes What is the probability of drawing 2 red socks? These are independent events R B G

4.5 Tree Diagrams and Outcome Tables Mr. Lieff is playing Texas Hold’Em He finds that he wins 70% of the pots when he does not bluff He also finds that he wins 50% of the pots when he does bluff If there is a 60% chance that Mr. Lieff will bluff on his next hand, what are his chances of winning the pot? We will start by creating a tree diagram

Tree Diagram 0.5 Win pot P=0.6 x 0.5 = 0.3 bluff 0.6 0.5 Lose pot 0.7 0.4 P=0.4 x 0.7 = 0.28 no bluff 0.3 Lose pot P=0.4 x 0.3 = 0.12

4.5 Tree Diagrams and Outcome Tables P(no bluff, win) = P(no bluff) x P(win | no bluff) = 0.4 x 0.7 = 0.28 P(bluff, win) = P(bluff) x P(win | bluff) = 0.6 x 0.5 = 0.30 Probability of a win: 0.28 + 0.30 = 0.58 So Mr. Lieff has a 58% chance of winning the next pot

Continued… P(no bluff, win) = P(no bluff) x P(win | no bluff) P(bluff, win) = P(bluff) x P(win | bluff) = 0.6 x 0.5 = 0.30 Probability of a win: 0.28 + 0.30 = 0.58 So Mr. Lieff has a 58% chance of winning the next pot

Review Read class notes and home learning Complete pp. 268-269 #3-4, 5abceg, 7, 9, 10 p. 270 #1-2, 6 Practice Test on web site