Advanced Math: – Lesson 10FGHI Compound Events, Venn Diagrams, and Expectation 10F / 4 10G / 2-­‐5 10H / 3-­‐6 10I / 1, 3, 6, 7 Warm-­‐Up 1) Both of the spinners show are spun. a) Draw a 2-­‐dimensional grid to illustrate the sample space. b) Determine the probability of getting: i) red and an odd number ii) red or an odd number iii) yellow and a number divisible by 3 iv) not yellow and a number divisible by 3. Compound Events _______________________ events are events that do not influence each other. Their probability can be found by: Ex1) Find the following probabilities and compare with your answers from Ex1. Determine the probability of: a) a red and an odd number b) not yellow and a number divisible by 3 Ex2) The probability of it raining on a weekend is 60% for Saturday and 70% for Sunday. Determine the probability of: a) it raining both Saturday and Sunday b) it raining Saturday, but not Sunday Ex3) A bag contains 5 blue marbles, 3 red marbles, 8 yellow marbles, and 4 green marbles. If you pick two marbles and replace each marble after picking it, what is the probability of picking: a) 2 blue marbles b) a red, then a green Ex4) The same experiment is repeated (5 blue, 3 red, 8 yellow, 4 green), but the marbles are NOT replace after each is picked. Find the probability of picking: a) 2 blue marbles b) a red, then a green Warm-­‐Up 2) Shade the following Venn Diagrams. a) both A and B b) either A or B c) not A U
U
U
A B A A B Warm-­‐Up 3) Use the values given to fill in the Venn Diagram. U
n(U) = 40 A n(A) = 24 d) only A or only B B U
A B B n(B) = 27 n( A' ∩ B' ) = 5 Ex5) A football team has 38 players. 20 are an only child, 7 are adopted, and 5 are both an only child and adopted. a) Place the information in the Venn Diagram alongside. O
A
b) Determine the number of players: i) who are an only child, but not adopted ii) neither adopted nor only child c) Determine the probability that a player chosen at random: i) is an only child ii) is adopted, but not an only child iii) is an adopted, only child Ex6) In a class of 17 students, 12 students have pet animals at home. Of those students, 7 have dogs at home. Let the event A represent having pets, and let the event D represent having dogs. a) Draw a Venn Diagram to represent the situation. b) Find P(A and D). c) Find P(A but not D). d) Find P(neither A nor D). Ex7) In a class of 30 students, 19 study Physics, 17 study Chemistry, and 15 study both of these subjects. a) Draw a Venn Diagram to represent the information. b) Find the probability of a student i) taking exactly one the subjects ii) taking at least one of the subjects iii) not taking Chemistry c) Let P represent the event of studying Physics and C the event of studying Chemistry. Find i) P(P and C) ii) P(P but not C) iii) P(P ∩ C)' Expectation If there are n trials and the probability of a single event occurring is p, then the expectation of the occurrence of that event is given by __________. Ex8) When an archer fires at a target, she hits the bullseye 2 out of every 5 shots. In a competition she is required to fire 40 arrows. How many times would you expect her to hit the bullseye? Ex9) In a game of chance, the player spins a square spinner labeled 1, 2, 3, 4, and wins the amount of money shown in the table below. Number 1 2 3 4 a) What is the expected return for one spin? Winnings $1 $2 $5 $8 b) Would you recommend a person to play this game if it cost $5 to play? Why? Ex10) Two dice are rolled simultaneously 180 times. On how many occasions would you expect to get a sum of 8? Ex11) In a coin game, a player flips a coin. If the coin lands on heads, the game is over. If the coin lands on tails, the player gets a point and flips the coin again. The coin may be flipped a maximum of three times before the turn is over. a) Fill in the tree-­‐diagram to represent the possibilities in one turn. Then complete the date of probabilities. Points
Outcomes
Probability
b) Find the expected return for playing the game. Ex12) You are asked to play the following game: You pay 2 CHF for the chance to roll a die. If you get an odd number, you win 1 franc. If you get a 2 or a 4, you lose 2 francs. If you get a 6, you win 10 francs. How much, on average, can you expect to win (or lose) in the long run?