Chapter 4 Review MDM 4U Mr. Lieff

4.1 Intro to Simulations and Theoretical Probability be able to design a simulation to investigate the experimental probability of some event ex: design a simulation to determine the experimental probability of more than one of 5 keyboards chosen in a class will be defective if we know that 25% are defective get a shuffled deck of cards, choosing clubs to represent the defective keyboards choose 5 cards and see how many are clubs repeat a number of times and calculate probability

4.2 Theoretical Probability work effectively with Venn diagrams ex: create a Venn diagram illustrating the sets of face cards and red cards S = 52 red & face = 6 face = 6 red = 20

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)/n(S) = 20/30 = 0.666 66.6%

4.2 Theoretical Probability ex: calculate the probability of not throwing a total of four with 3 dice there are 63 possible outcomes with three dice only 3 outcomes produce a 4 probability of a 4 is: 3/63 probability of not 4 is: 1- 3/63

4.3 Finding Probability Using Sets recognize the different types of sets utilize the additive principle for unions of sets 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) calculate probabilities using the additive principle

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

4.4 Conditional Probability calculate a probability of events A and B occurring, given that A has occurred use the multiplicative law for conditional probability ex: what is the probability of drawing a jack and a queen in sequence, given no replacement? 4/52 x 4/51

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 a) Draw a Venn Diagram that represents this situation. b) What is the probability that a student takes Mathematics given that he or she also takes English?

4.4 Conditional Probability E 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 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 n(S) 43

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. Greer is going fishing he finds that he catches fish 70% of the time when the wind is out of the east he also finds that he catches fish 50% of the time when the wind is out of the west if there is a 60% chance of a west wind today, what are his chances of having fish for dinner? we will start by creating a tree diagram

4.5 Tree Diagrams and Outcome Tables 0.5 fish dinner P=0.3 west 0.6 0.5 bean dinner P=0.3 fish dinner 0.7 P=0.28 0.4 east 0.3 bean dinner P=0.12

4.5 Tree Diagrams and Outcome Tables P(east, catch) = P(east) x P(catch | east) = 0.4 x 0.7 = 0.28 P(west, catch) = P(west) x P(catch | west) = 0.6 x 0.5 = 0.30 Probability of a fish dinner: 0.28 + 0.3 = 0.58 So Mr. Greer has a 58% chance of catching a fish for dinner

4.6 Permutations find the number of outcomes given a situation where order matters calculate the probability of an outcome or outcomes in situations where order matters recognizing how to restrict the calculations when some elements are the same

4.6 Permutations ex: in a class of 10 people, a teacher must pick 3 for an experiment (students are done in a particular order) how many ways are there to do this? ans: P(10,3) = 10!/(10 – 3)! = 720 ex: how many ways can 5 students be arranged in a line? ans: 5! ex: how many ways are there above if Jake must be first? ans: (5-1)! = 4!

4.6 Permutations ex: what is the chance of opening one of the school combination locks by chance? ans: 1 in 60 x 60 x 60 = 1 in 216 000

4.7 Combinations find the number of outcomes given a situation where order does not matter calculate the probability of an outcome or outcomes in situations where order does not matter ex: how many ways are there to choose a 3 person committee from a class of 20? ans: C(20,3) = 20!/((20-3)!3!) = 1140

4.7 Combinations ex: from a group of 5 men and 4 women, how many committees of 5 can be formed with a. exactly 3 women b. at least 3 women ans a: ans b:

Combinatorics (§4.6 & 4.7) Permutations – order matters E.g. President Combinations – order does not matter E.g. Committee

Test Review pp. 268-269 #1, 4, 5-6 a,c,e…, 8, 9, 11 p. 270 #1, 2