1. Math 30-2 Probability & Odds

3. Acceptable Standards (50-79%)  The student can express odds for or odds against as a probability determine the probability of an event determine odds for or odds against an event determine the odds for an event given the odds against the event and vice versa distinguish between mutually exclusive events and non-mutually exclusive events determine P(A or B) for events that are mutually exclusive determine P(A) when given P(A or B) and P(B) for mutually exclusive events interpret a model that represents any combination of mutually exclusive and non-mutually exclusive events identify events that are complementary describe the elements that belong to the complement of a simple event determine the probability of the complement of an event, given the probability of the event create a sample space using a graphic organizer distinguish between independent and dependent events determine P(A and B) for independent events determine P(A and B) for dependent events, given the order of the events

4. Standards of Excellence (80% +)  The student can also express probability as odds for or odds against provide an explanation for the validity of a probability statement provide an explanation for the validity of an odds statement determine P(A or B) for events that are non-mutually exclusive determine P(A) when given P(A or B), P(A and B), and P(B) for non-mutually exclusive events represent events that are non-mutually exclusive using a graphic organizer describe the elements that belong to the complement of a compound event determine P(A) when given P(A and B) and P(B) for independent events determine P(A and B) for dependent events when the order of the events is not given

5. Vocabulary  Probability (P) – is the likelihood that an event will occur.  Outcomes – when you do a probability experiment the different possible results are called outcomes  Event – is a collection of outcomes  Sample space: the set of all possible outcomes. We denote S

6. Listing the Sample Space Use a tree diagram to list the sample space for tossing a coin and rolling a die. CoinDieOutcomes H 1 2 3 4 5 6 1 2 3 4 5 6 T H, 1 H, 2 H, 3 H, 4 H, 5 H, 6 T, 1 T, 2 T, 3 T, 4 T, 5 T, 6 The Sample Space

7. Types of Probability  There are 2 types of probability Theoretical Probability Experimental Probability  Let ’ s look at each one individually…

8. Theoretical Probability  Theoretical Probability is based upon the number of favorable outcomes divided by the total number of outcomes Example: In the roll of a die, the probability of getting an even number is 3/6 or ½. Theoretical Probability Formula:

9. Example # 2  A box contains 5 green pens, 3 blue pens, 8 black pens and 4 red pens. A pen is picked at random  What is the probability that the pen is green? There are 5 + 3 + 8 + 4 or 20 pens in the box P (green) = # green pens = 5 = 1 Total # of pens 20 4

10. Experimental Probability  As the name suggests, Experimental Probability is based upon repetitions of an actual experiment. Example: If you toss a coin 10 times and record that the number of times the result was 8 heads, then the experimental probability was 8/10 or 4/5 Experimental Probability Formula: P = Number of favorable outcomes Total number trials

11. Example  In an experiment a coin is tossed 15 times. The recorded outcomes were: 6 heads and 9 tails. What was the experimental probability of the coin being heads? P (heads) = # Heads = 6 Total # Tosses 15

13. The sum of all the probabilities of an event is equal to 1. If P = 1, then the event is a certainty. If P = 0, then the event is impossible. In probability, if Event A occurs, there is also the probability that Event A will not occur. Event A not occurring is the compliment of Event A occurring. The probability of Event A not occurring is written as P(A). (This is read as “Probability of not A”). For Event A: P(A) + P(A) = 1 P(A) = 1 - P(A) Complementary

14. One card is drawn from a deck of 52 cards. What is the probability of each of these events? a) drawing a red four b) not drawing a red four Example

15. Odds

16.  Another way to describe the chance of an event occurring is with odds. The odds in favor of an event is the ratio that compares the number of ways the event can occur to the number of ways the event cannot occur.  We can determine odds using the following ratios: Odds in Favor = number of successes number of failures Odds against = number of failures number of successes Also can write it as: odds in favor of A = number of outcomes for A : number of outcomes against A

17. Example  Suppose we play a game with 2 number cubes. If the sum of the numbers rolled is 6 or less – you win! If the sum of the numbers rolled is not 6 or less – you lose

18. In this situation we can express odds as follows: Odds in favor = numbers rolled is 6 or less numbers rolled is not 6 or less Odds against = numbers rolled is not 6 or less numbers rolled is 6 or less

19. Example #2  A bag contains 5 yellow marbles, 3 white marbles, and 1 black marble. What are the odds drawing a white marble from the bag? Odds in favor = number of white marbles3 number of non-white marbles6 Odds against = number of non-white marbles6 number of white marbles3 Therefore, the odds for are 1:2 and the odds against are 2:1

21. Practice Suppose you are watching a game show on TV. Five green doors are shown. Contestants on the show get to choose a door and potentially win a prize. Prizes can be found behind two of the five doors. 1) Determine the probability of winning a prize. 2) Determine the probability of not winning a prize. 3) Add the probability of winning and not winning a prize. What do you notice? 4) Use the following formula to write the odds as a fraction. 5) Write the odds of winning as a ratio. odds in favour of A = number of outcomes for A : number of outcomes against 6) Write the odds against winning as a ratio.

22. Discussion  How does the probability of winning a prize compare to the odds of winning a prize?  What similarities and differences do you notice between expressing the probability and the odds for an event?  What do you notice about the odds for winning versus the odds against winning? What relationship do you see?

23. Practice 2 There is a 10% probability of winning a free play in the charity draw. 1) Write 10% as a fraction. 2) If 100 tickets are purchased, theoretically how many tickets would win a free play? 3) If 100 tickets are purchased, theoretically how many tickets would not win a free play? 4) Based on a 100 tickets being sold, what are the odds in favour of winning a free play? Write your answer as a ratio; then write it as a reduced ratio. 5) Describe in words the information you can gain from knowing the probability of an event and how this information helps you write the odds for the event.

25. Probability vs Odds  Probability is based on winning a prize out of all of the possibilities, whereas odds are based on winning a prize compared to not winning a prize.  The difference between odds and probability is this: Probability is based on favourable outcomes in relation to the total number of possible outcomes. Odds are based on the favourable outcomes “for” in relation to unfavourable outcomes “against.”

27. Classifying Exclusivity Two events are mutually exclusive if they cannot occur simultaneously. For instance, the events of drawing a diamond and drawing a club from a deck of cards are mutually exclusive because they cannot both occur at the same time. For mutually exclusive events: P(A B) = P(A) + P(B) Mutually Exclusive: AB diamond club

28. Classifying Exclusivity Events that are not mutually exclusive have some common outcomes. For instance, the events of drawing a diamond and drawing a king from a deck of cards are not mutually exclusive because the king of diamonds could be drawn, thereby having both events occur at the same time. For events that are not mutually exclusive: P(A B) = P(A) + P(B) - P(A and B) Non- Mutually Exclusive

29. Venn Diagram: Events that are Not Mutually Exclusive King Diamonds Both king and a diamond These events are not mutually exclusive as it is possible for a card to be both king and a diamond

30. Classify each event as mutually exclusive or not mutually exclusive. a) choosing an even number and choosing a prime number b) picking a red marble and picking a green marble c) living in Edmonton and living in Alberta d) scoring a goal in hockey and winning the game e) having blue eyes and black hair Classifying Exclusivity

31. 1. A box contains six green marbles, four white marbles, nine red marbles, and five black marbles. If you pick one marble at a time, find the probability of picking a) a green or a black marble. b) a white or a red marble. Example

32. Determine the probability of choosing a diamond or a face card from a deck of cards. Example 2

33. A national survey revealed that 12.0% of people exercise regularly, 4.6% diet regularly, and 3.5% both exercise and diet regularly. What is the probability that a randomly-selected person neither exercises nor diets regularly? Example 3

34. Work on Practice Questions 1-8

35. P(A and B) = P(A) x P(B) (INDEPENDENT EVENTS) Independent Versus Dependent Events Two events are independent if the probability that each event will occur is not affected by the occurrence of the other event. If the probabilities of two events are P(A) and P(B) respectively, then the probability that both events will occur, P(A and B), is:

36. If A and B are events from an experiment, the conditional Probability is the probability that Event B will occur given that Event A has already occurred. (dependent event) Conditional Probability P(B|A) is the notation for conditional probability. It should be read as “the probability of event B happening, given that event A has already occurred.” A tree diagram is useful for modeling this types of problems

37. Practice Two events are dependent if the outcome of the second event is affected by the occurrence of the first event. Classify the following events as independent or dependent: a) tossing a head and rolling a six b) drawing a face card, and not returning it to the deck, and then drawing another face card c) drawing a face card and returning it to the deck, and then drawing another face card

38. A cookie jar contains 10 chocolate and 8 vanilla cookies. If the first cookie drawn is replaced, find the probability of: a) drawing a vanilla and then a chocolate cookie b) drawing two chocolate cookies Example

39. Find the probability of drawing a vanilla and then drawing a chocolate cookie, if the first cookie drawn is eaten. Example

40. Determine the conditional probability for each of the following: a) Given P(A and B) = 0.725 and P(A) = 0.78, find P(B|A). b) Given P(blonde and tall) = 0.5 and P(B|A) = 0.68, find the P(blonde). Practice

41. The local hockey time is having a raffle to raise money. The team is selling 2500 tickets, and there will be two draws. The first draw is for the grand Prize—a trip for two to an all-inclusive resort. The second draw is for the consolation prize-an HDTV. After each draw, the winning ticket is not return to the raffle. You buy 10 tickets for the raffle. Calculate the probability of winning the HDTV. What is the probability of winning at least one prize? Finding Conditional Probability

43. A diagnostic test for liver disease is accurate 93% of the time, and 0.9% of the population actually has liver disease. a) Determine the probability the patient tests positive b) Determine the probability the patient tests negative c) Determine the probability the patient has liver disease and tests positive d) Determine the probability the patient does not have liver disease and tests negative