The likelihood of an event occurring is not always expressed in terms of probability.

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1 Lesson #5 Probability and Odds The likelihood of an event occurring is not always expressed in terms of probability. The likelihood of an event occurring can also be expressed in terms of the odds in favor of it occurring. Odds in favor = favorable outcomes : unfavorable outcomes Odds against = unfavorable outcomes : favorable outcomes The probability of an event occurring and the odds in favor of it occurring are not the same thing. Example 1 The odds in favor of an event occurring are 3:1. Determine the probability of the event occurring. Solution: The odds of the event occurring are 3:1, so the number of ways the event can occur is and the number of ways the event cannot occur is. This means that the total number of possible events is. The probability of the event occurring is. Example 2 Three coins are tossed, determine: a. The probability of 3 coins all landing heads. b. The odds in favor of 3 coins all landing heads. c. The odds against 3 coins all landing heads. Page 1 of 9

2 Example 3 Silas is buying new television televisions did not need repairs over a 3 year period and 1000 television did. a. What are the odds in favor of needing repairs? b. What are the odds against needing repairs? c. What is the probability that the television will need repairs during the 3 year warranty. Page 2 of 9

3 Assignment #5 Probability and Odds 1. The odds against a hockey team winning a game are 2 : 9. What is the probability that the team will win a game? 2. Nathan plays basketball. He has scored on 2 out of 10 shots. He says that his odds against scoring are 4 to 1. Do you agree? Explain. 3. Each letter of the word MATHEMATICAL is written on a different card and placed face down on a table. a. Determine the probability of drawing an M. b. Determine the odds in favor of drawing an M. c. Determine the probability of not drawing an M. d. Determine the odds against drawing an M. 4. Susan works at an appliance store in Brandon. The odds that a new vacuum cleaner will need repairs in the first 4 years are 1 : 3. a) What is the probability that a new vacuum cleaner will need repairs? b) Is this a good vacuum cleaner in your opinion? c) If the vacuum cleaner cost $350 and the manufacturer gave a 4 year warranty that cost $50, would you buy the warranty? Why or why not? Page 3 of 9

4 5. A die is rolled. Find the following: a. The probability of rolling a number greater than two. b. The odds in favor of rolling a number greater than two. c. The probability of not rolling an even number. d. The odds against rolling an even number. 6. In a class of 32 students, 18 students take an Art option, 10 other students take a Drama option, and the rest of the students take a Choir option. One student is selected at random. Find the following: a. The odds in favor of the selected student taking Drama. b. The odds against the selected student taking Choir. c. The odds in favor of the selected student either taking Art or Choir. 7. The Health Science Center Lottery states that players have a 1 in 12 chance of winning something in their lottery. What are the odds in favor and the odds against winning this lottery? 8. In roulette, the odds of winning on a single number is 1 : 37. What is the probability of winning in this game? What is the probability of not winning? Page 4 of 9

5 Lesson #6 Expected Value Expected value is an application of probability which involves the likelihood of a gain or a loss. Expected value is relevant in business, insurance, and many situations in our daily life. To determine the expected value of something, first you find the probability of the required event, and then you find the loss or gain associated with that event. You then multiply these together. Example: Consider the following game. You have a 1 in 5 chance of winning the game and a 4 in 5 chance of losing the game. The game costs $1 to play each time. If you win the game, you receive $4 and if you lose the game, you receive nothing. Find the expected value of the game. Solution: There are two events: winning the game or losing the game. The probability of winning the game is 1/5 and the probability of losing the game is 4/5. P(win) = probability of winning = $gain = amount won cost of playing = P(lose) = probability of losing = $loss = cost of playing = Expected Value = Page 5 of 9

6 Assignment #6 Expected Value 1. Mr. Krahn is at the Red River Ex. He wants to play the ring toss and win a stuffed animal for his wife. It will cost $3 each time he plays the game. He has a 1 in 10 chance of winning the $25 stuffed bear, a 1 in 5 chance of winning $5 stuffed duck, and a 7 in 10 chance of winning nothing. Fill in the following chart below. Event Win (stuffed bear) Win (stuffed duck) Lose (win nothing) Probability Amount Won Cost of Playing Payoff Probability X Payoff What is the expected value of the game? Remember: $gain = amount won cost of playing Page 6 of 9

7 2. Aaron is thinking about adding collision insurance when he renews his carinsurance policy next year. This will increase the cost of his insurance by $720 per year. Statistically, there is a 99.4% chance that Aaron will not have a collision in the next year. If Aaron has a collision, the insurance company will pay $5000. a. What is the probability that Aaron will have a collision? b. What is the expected value for the company if Aaron adds collision insurance? c. Should Aaron add collision insurance? Explain. Page 7 of 9

8 3. A building contractor sets her probability of winning a contract at.30. The contract is worth $ and she determines it will cost her $2400 to prepare a contract proposal. a. Find the expected value of the contract proposal. b. Is it a good financial decision for her to bid on the contract? Why? c. What other factors might she consider before deciding whether to bid on the contract? Page 8 of 9

9 4. Dan is buying a clothes dryer that costs $825. The clothes dryer has a 1 yr warranty from the manufacturer and an extended 5 yr warranty that cost $50. The odds against needing repairs over 5 yrs are 3:17. a. What is the probability that the clothes dryer will need repairs during the extended warranty period? b. What is the probability that the dryer will not need repairs. c. The company estimates that it costs $200 to repair a dryer. Suppose that Dan buys the warranty. What is the expected value for the store? d. Should Dan buy the warranty? Explain. What assumptions did you make? Page 9 of 9