Probability Topic 2: Probability and Odds

I can explain, using examples, the relationship between odds and probability. I can express odds as a probability and vice versa. I can determine the odds for and against an outcome in a situation. I can explain, using examples, how decisions may be based on odds. I can solve a contextual problem that involves odds.

Explore… Suppose that, at the beginning of a regular CFL season, the Edmonton Eskimos are given a 25% chance of winning the Grey Cup. 1. What is the event in the situation described above? 2. Express the probability that this event will occur as a fraction out of 100. Try the Explore on your own first. Then look at the solutions. Edmonton Eskimos winning the Grey Cup.

Explore… 3. Describe the complement of this event. 4. Express the probability that the complement will occur as a fraction out of 100. Edmonton Eskimos not winning the Grey Cup. If the probability of the Eskimos winning is 25%, the probability of them not winning is 75%.

Explore… 5. Write the odds in favour of the Eskimos winning the Grey Cup. 6. Write the odds against the Eskimos winning the Grey Cup. The odds in favour is the ratio of it occurring (25) to the ratio of it not occurring (75). 25:75 The odds against is the ratio of the probability of an event not occurring (75) to the probability of it occurring (25). 75:25

Explore… 7. How are the odds in favour of them winning related to the odds against them winning? 8. How does the probability of an event and the probability of its complement relate to the odds in favour of the event occurring? They contain the same numbers, reversed in order. Odds in favor is made up of the numerators of the probabilities of the event and its complement. 25:75

Explore… 9. How are the odds in favour of an event similar to its probability? 10.How are the odds in favour of an event different from its probability? Odds in favour is favourable to unfavourable outcomes. Probability is favourable to total outcomes.

Information In Set Theory, we learned about the complement. The complement can also be used when dealing with probability. The sum of the probability of an event and its complement always equals 1. So we can say: Odds express a level of confidence about the occurrence of an event. It is a way to compare the probability of something happening to the probability of it not happening.

Information Odds can be expressed in two ways: Odds in Favour ratio of the probability that an event will occur to the probability that it will not occur: ratio of the number of favourable outcomes to the number of unfavourable outcomes: Odds Against ratio of the probability that an event will not occur to the probability that it will occur: ratio of the number of unfavourable outcomes to the number of favourable outcomes:

Example 1 Suppose that the probability of an event happening is. a) Calculate the complement of this event. b) What are the odds in favour of the event happening? c) What are the odds against the event happening? Finding odds from probability

Example 2 Consider a standard deck of 52 playing cards. Bailey holds all the hearts. He asks Morgan to choose a single card without looking. Finding Odds using Sets

Example 2 a)Define the universal set, H, of all hearts. b) Define the set, F, of all hearts that are face cards. c) Define the set, F’, of all hearts that are not face cards. Finding Odds using Sets H = {1H, 2H, 3H, 4H, 5H, 6H, 7H, 8H, 9H, 10H, JH, QH, KH} F = {JH, QH, KH} F’ = {1H, 2H, 3H, 4H, 5H, 6H, 7H, 8H, 9H, 10H}

Example 2 d)Find the odds in favour, n(F):n(F’), of Morgan choosing a heart face card. e) Find the odds against, n(F’): n(F), of Morgan choosing a heart face card. f) Do you think Morgan is more likely to draw a face card or something different? Explain. Finding Odds using Sets 3:10, since there are 3 hearts that are face cards and 10 hearts that are not. 10:3, since there are 10 hearts that are not face cards and 3 hearts that are. Morgan is more likely to draw a non-face card than a face card since there are more of them.

Example 3 An oil and vinegar salad dressing is made using 2 parts oil to 1 part vinegar. So, the ratio of oil:vinegar is 2:1. What fraction of the dressing is oil? Converting odds to a fraction There are 2 parts oil and 1 part vinegar, so 3 parts in total.

Example 4 A computer randomly selects a university student’s name from the university database to award a $100 gift certificate for the bookstore. The odds against the selected student being male are 57:43. Determine the probability that the randomly selected university student will be male. Finding probability from odds The question says that the odds against the selected student being male is 57:43. This means that there is a ratio of 57 women to 43 men.

Example 5 Suppose the odds in favour of Jack winning a race are 5:3. Suppose the odds in favour of Bill winning the same race are 6:4. Determine who has a better chance of winning the race. Comparing odds in favour

Example 5 Comparing odds in favour ii. You can use equivalent ratios Jack’s odds in favour is 5:3. Bill’s odds in favour is 6:4. These ratios can be changed to equivalent ratios by finding the lowest common multiple of 3 and 4, which is 12. By multiplying both terms in Jack’s ratio by 4, we can determine an equivalent ratio of 20:12. By multiplying both terms in Bill’s ratio by 3, we can determine an equivalent ratio of 18:12. Jack’s odds in of winning (20:12) are higher than Bill’s odds of winning (18:12).

Example 6 A hockey game has ended in a tie after a 5 minute overtime period, so the winner will be decided by a shootout. The coach must decide whether Leila or Caris should go first in the shootout. The coach would prefer to use her best scorer first, so she will base her decision on the players’ shootout records. Based on odds in favour, who should go first? Making a decision based on odds in favor PlayerGoals ScoredGoals Missed Leila85 Caris107 Leila’s odds in favour is 8:5. Caris’s odds in favour is 10:7. These ratios can be changed to equivalent ratios by finding the lowest common multiple of 5 and 7, which is 35.

Example 6 Making a decision based on odds in favor Both terms in Leila’s odds in favour (8:5) can be multiplied by 7 to determine the equivalent ratio 56:35. Both terms in Caris’s odds in favour (10:7) can be multiplied by 5 to determine the equivalent ratio 50:35. These ratios can be changed to equivalent ratios by finding the lowest common multiple of 5 and 7, which is 35. Leila has better odds in scoring, so she should go first.

Need to Know Odds express a level of confidence about the occurrence of an event. It is a way to compare the probability of something happening to it not happening. The sum of any event and its complement is 1 or 100%. D J

Need to Know Odds can be determined from both probability and sets: Odds in FavourOdds Against From Probability From Sets

Need to Know If the odds in favour of event A occurring are m : n, then the odds against event A occurring are n : m. Probability can be determined from odds: If the odds in favour of event A occurring are m : n, then If the odds against event A occurring are n : m, then When comparing two odds, you may interpret them better by converting the odds to equivalent ratios or to probability. You’re ready! Try the homework from this section.