Probability Essential Math 12 Mr. Morin Name: Slot: Introduction Probability and Odds Single Event Probability and Odds Two and Multiple Event Experimental and Theoretical Probability Expected Value (Expected Win/Loss) Probability in the Workplace 1
Essential Math 12 Probability Introduction Fill in the table. Percent Fraction Decimal Words 13% 13 0.13 13 out of 100 100 3 4 0.12 4% 12 25.34 Calculate the following 12% of 20 10% of 42 24% of 48 20% of 113 113% of 25 50% of 80 30% of 40 55% of 88 Probability and Odds 2
Example 1 A bag has six balls in it. The colours are as follows: red, green, yellow, orange, black, and white. One ball will be drawn from the bag. Identify the event: Identify all possible outcomes: Example 2 Bob is drawing marbles from a bag: there are 5 green, 3 yellow and 1 red. a) State the event. b) State all the possible outcomes; how many possible outcomes. c) State the probability of drawing a yellow. d) State the odds for drawing green. e) State the odds against drawing red. f) What is the probability of drawing green or red? Questions - Probability and Odds 1) Given the event, identify all possible outcomes Rolling a six-sided cube Flipping a coin Event: Outcomes: Event: Outcomes: Spin a spinner with four colours: red, orange, blue, and green. Event: Outcomes: 3
2) If you roll a six-sided cube (die) a) Calculate the probability of rolling a 2. b) Calculate the probability of rolling an even number. c) Calculate the probability of rolling a number greater than 4. d) Calculate the probability of not rolling a 6. 3) A spinner has four colours: red, orange, blue, and green. a) Calculate the probability of not spinning blue. b) State the odds for spinning red. c) State the odds against spinning red. d) State the odds for spinning green or blue. e) State the odds against spinning green or blue. 4) The weather forecast states that there is a 30% probability of rain tomorrow. a) State the odds against it raining tomorrow. b) State the odds for it raining tomorrow. 4
5) There are ten numbers in a hat numbered 1through 10. A ball is drawn at random. a) Probability of picking an even number b) State the odds for picking a 7. c) State the odds against drawn a number less than 3. d) State the odds against drawing a number more than 4. e) State the odds for drawing a four. 6) The odds of snow this Tuesday is 5:4. Express the probability of it not raining as a) Decimal b) Percent c) Fraction 7) Identify a situation where the odds are 5:2. Odds and Probability Two Event & Multiple Event 5
Example 1 A six-side cube (die) is rolled and a spinner with the colours: orange, yellow, brown and grey is spun at the same time. State the event and their individual outcomes Event 1: Event 2: Outcomes for Event 1: Outcomes for Event 2: Make a probability tree to list all possible outcomes. Example 2 There are 4 balls in a hat numbered 1through 4. A ball is drawn and returned to the hat and then second ball is drawn. Make a probability tree and a matrix to make a list of all outcomes. 6
Questions: Two Event & Multiple Event 1) A coin is flipped twice. State Event 1: Outcomes of Event 1: Event 2: Outcomes of Event 2: Draw a probability tree for this situation. State all possible outcomes for this situation. 2) A spinner with colours red, blue, black and white is spun twice. Event 1: Outcomes of Event 1: Event 2: Outcomes of Event 2: a) Draw a probability tree for this situation. b) State all possible outcomes. c) State the probability of spinning blue, blue? P(B,B) 7
d) State the probability of not spinning red, red? e) State the odds of spinning one blue and one red (order does not matter). 2) A coin is flipped twice. a) State the probability of flipping heads, heads. b) State the probability of flipping one head and one tail. 3) Two dice are rolled. a) State the probability of rolling 1 and 1 (snake eyes). b) State the odds against rolling a sum less than 5. 8
4) April and Ryan will have four children. a) What is the probability that they will have at least 3 girls? b) What are the odds for having 4 children of the same gender? 9
Experimental and Theoretical Probability Example 1 A die is rolled 542 times and a 6 occurs 151 times. a) State the theoretical probability of rolling a 6 b) State the experiment probability. c) Is it possible that 6 is rolled 12 consecutive times? Questions - Experimental and Theoretical Probability 1) A coin is flipped 974 times and it has been heads 532 times. a) State the theoretical probability of flipping a heads. b) State the experimental probability of flipping a tails. 10
2) You are roll a six-sided cube (die) 7 times. The outcomes of the six rolls are as follows: 5, 1, 3, 5, 2, 5, 5. a) State the experiment probability. b) State the theoretical probability. 3) The probability of being left handed is approximately 1 in 11. Out of 13,250 how many people to you expect to have left handed. 4) A cup from Tim Horton s Roll up the Rim to Win contest being a winner is approximately 1 out 9. Jessica buys 9 cups of coffee and did not get a winner. She insists that the statement 1 out of every 9 cups is a winner must be wrong. Is 1 out of every 9 cups correct? Please state your reasoning. 5) You flip a coin five times and the following occurred: heads, heads, tails, heads, heads. a) State the experimental probability. b) What is the probability flipping a heads on the next flip. Explain. 6) A quality control specialist at a factory knows that 2% of widgets are defective. On a day he notices that there were 20 of 200 widget made were defective. Should the quality control specialist be concerned that too many widgets are defective? Explain. 11
7) Erwin is a farmer in rural Manitoba. There is an equal probability that a farmer in Erwin s area will plant one of two crops: wheat or canola. Erwin surveys 10 farmers in the area and finds out that 7 of them plan to plant wheat. a) State the theoretical probability that a surveyed farmer will plant wheat. b) State the experimental probability that a surveyed farmer will plant canola. c) Explain why Erwin might decide to plant canola even though most farmers in the area are planning to plant wheat. 8) State an example where having an item failing 10% of the time is acceptable. 9) State an example where having an item failing 0.001% unacceptable. 12
Expected Value (Expected Win/Loss) Example 1 A card game has five cards, one of which is the winner. The game costs $1.00 to play. If you select the correct card, you win $4.00. Calculate the expected win/loss. Interpret the meaning of this number. Example 2 Rachel owns a window cleaning company. She is considering putting in a bid for a contract. She calculates the expected value to be $425. Interpret the meaning of the expected value. Should she bid on the contract? 13
Questions - Expected Value (Expected Win/Loss) 1) Based on past experience, a building contractor estimates that the probability of winning a contract is 0.3. The contract is worth $25,000 and she knows it will cost her $2,400 to prepare a contract proposal. Find the expected win/loss of the contract proposal. Is it financially a good idea for her to bid on the contract? 2) For each of the following games determine whether you would win, lose, or break even, if played many times. a. Pay $1. Toss a coin. If it shows heads, you win $2. b. Pay $1. Draw a card from a shuffled deck. If it is a heart you get $5. c. Pay $2. Draw a card from a shuffled deck. If it is a jack or an ace, you get $10. d. Pay $1, Roll a die. If a 2 or 3 shows, you get $4. e. Pay $1. Toss two coins. If they are both heads, you win $3. 14
3) Based on past experience, a systems engineer sets the probability of winning a computer contract at 0.25. The contract is worth $10,000 and the engineer calculates it would cost $1,800 to prepare a contract proposal. a) Find the expected value of the contract proposal. How much will the engineer gain if he wins the contract? b) Is it financially a good idea for the engineer to bid on the contract? Explain. 4) A carpet cleaning company uses flyers to promote its business. The company knows that on average 1 out of every 100 households receiving its flyer will use its service. One average, the company makes a profit of $50.00 on each household whose carpets it cleans. It costs the company $0.25 for each flyer. a. Find the expected value of each flyer. b. Find the expected value of 10 000 flyers. c. Is it financially a good idea for the company to send out flyers? Please explain 15
5) A community club raffles off a $1500 big screen TV. The tickets each cost $5 and there are 2500 tickets sold. a) What is your expected value if you had bought only one of the tickets? b) How does the expected value change if you buy five of these tickets? Is it better to purchase more tickets? 6) A manufacturer builds and sells widgets. He know that, on average, 13 widgets per 100 are defective. It costs $12 to build each widget and they sell for $18 each. He plans on building 100 000 this year. Will the manufacturer make or lose money on widgets this year? 16
Probability in the Workplace 1) Mr. Morin has 48 students in Essential Math 12. When he is marking, five of the first six test all received a mark higher than 90%. Did he make the test too easy? Please provide rationale. 2) Paul is building a fence and needs 350 usable boards. At the lumber store the probability that a board will be useable is 85%. How many boards should Paul buy? Please explain. 3) Nancy is an excellent forklift operator. She has 33 years experience without an accident: her record is flawless. Amy, Nancy s boss, states the probability that Nancy will get into an accident is 0. Evaluate this statement. 4) Identify when landscaping companies and golf courses use probability in its operation. Please provide rationale. 17
5) At the casino, a roulette dealer noticed that the number 22 has occurred 3 times in a row. He expresses great concern to his boss. Was his concern justified? Explain. 6) Identify when Manitoba Public Insurance uses probability in its business. 18