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Foundations 30Unit 3 – Probability FP 30.4: Extend understanding of odds and probability.FP30.5: Extend understanding of the probability of two events, including events that are: mutually exclusive non-mutually exclusive dependent independent.. * Adapted from Nelson Foundations of MathematicsKey Terms Fair Game – A game in which all the players are equally likely to win; for example, tossing a coin to get heads or tails is a fair game. Experimental Probability – PA= n(A)n(T), where n(A) is the number of times event A occurred and n(T) is the total number of trials, T, in the experiment. Theoretical Probability – PA= n(A)n(S), where n(A) is the number of favourable outcomes for event A and n(S) is the total number of outcomes in the sample space, S, where all outcomes are equally likely. Odds in Favour – The ratio of the probability that an event will occur to the probability that the event will not occur. It could also be the ratio of the number of favourable outcomes to the number of unfavourable outcomes. Odds against – The ratio of the probability that and event will not occur to the probability that the event will occur. It could also be the ratio of the number of unfavourable outcomes to the number of favourable outcomes. Principle of Inclusion and Exclusion – The number of elements in the union of two sets is equal to the sum of the number of elements in each set, less the number of elements in both sets; using set notation this is written: nA∪B=nA+nB-n(A∩B)Mutually Exclusive – Two or more events that cannot occur at the same time; for example the sun setting and the sun is rising are mutually exclusive.Non-Mutually Exclusive – Two events, A and B, that are intersecting sets.Dependent Events – Events whose outcomes are affected by each other; for example, if two cards are drawn from a deck without replacement, the outcome of the second event depends on the outcome of the first event (first card).Conditional Probability – The probability of an event occurring given that another event has already occurred. Conditional Probability Notation – P(B|A) is read: The probability that event B will occur, given that event A has already occurred. Lesson 1 – Experimental and Theoretical ProbabilityExperimental Probability – PA= n(A)n(T), where n(A) is the number of times event A occurred and n(T) is the total number of trials, T, in the experiment. Different Ways to express Probability PercentFractionDecimalStatementDetermining Probability 1) Determine the total number of possible outcomes2) Determine how likely the event is3) Use the formula to express in whichever form you needExample 1: Shelley works at a store that sells light bulbs and finds that the current shipment has a high number of defective light bulbs. Shelley decides to test how probable it is that light bulbs from the assembly line are defect. She has workers on the assembly line randomly choose light bulbs from the assembly line and test them. Out of 265 light bulbs tested in the random sample, 2 were defective. Express the probability of a light bulb being defective as a decimal, a percentage, a fraction and in a statement.In a shipment of 5000, how many light bulbs are likely to be defective. Theoretical Probability – PA= n(A)n(S), where n(A) is the number of favourable outcomes for event A and n(S) is the total number of outcomes in the sample space, S, where all outcomes are equally likely. Calculating Theoretical Probability1) Determine the number of possible events or the number of not possible events2) Determine the total number of possible events3) Write as a probabilityExample 2: The following is a spinner with different colours.192405234315GGOOYYRRBB00GGOOYYRRBBDetermine the probability that the spinner will stop on green.Determine the probability that the spinner will not stop on yellow.Determine the probability that the spinner will stop on a red or an orange. Multiple Events1) Determine the probability of the first event2) Determine the probability of the multiple events after3) Multiply the probability togetherExample 3: A bag of finishing nails contains 5 nails measure 1”, 5 nails measuring 2” and 7 nails measuring 3”. A carpenter randomly chooses one nail and does not replace the nail. He then takes a second nail. What is the probability that the first nail will be 1”?What is the probability that the second nail, after the first nail has been chosen, will be a 3” nail?What is the probability that he will choose a 1” nail and then a 3” nail?Example 4: You are rolling a dice in a board game and need to make it to a certain square. You have two turns to get to this square. Your two rolls you need to get a 5 and a 6. a) What is the probability that you will roll a 5 your first roll?b) What is the probability that you will roll a 6 on your second roll? c) What is the probability that you will roll a 5 and then a 6? Example 5: You are drawing cards from a standard deck of 52 cards. You draw the first card, replace the card back in the deck, and pick again. What is the probability of drawing both cards to be a club?You draw the first card then keep the card out of the deck. What is the probability of drawing both cards to be hearts?You draw a first card to be a 5. What is the probability of drawing the next two cards, without replacing any cards on the deck, to be both cards higher than 5?Lesson 2 – OddsOdds in Favour – The ratio of the probability that an event will occur to the probability that the event will not occur.Odds Against – The ratio of the probability that an event will not occur to the probability that the event will occur. Example 1: A television station is hosting a contest. Alice, a participant, can choose 1 of 7 doors to open. She will win whatever prize is behind the door. Behind the 7 possible doors are the following prizes: One offers a trip;Two contain $20.00 gift certificates Three award restaurant dinners;One door has no prizea) What are the odds of choosing the door with no prize?b) What is the probability of choosing the door with no prize?c) What are the odds of Alice winning a gift certificate?Example 2: A tire manufacturing company does random testing of tires coming off the production lines to ensure that they are being produced correctly. After testing, the quality control manager calculates that the experimental probability of a tire having a defect is 0.0003a) What are the odds in favour of a tire being defective?b) What re the odds against a tire being defective?c) In a production line of 30 000 tires, how many would you expect to be defective and non-defective?Determining Sets for Odds and Probability1) Determine any sets and subsets2) Write out the elements of each set3) Determine the number of each set by counting each element Example 3: Determine Sets, subsets, and the number of each set in a standard deck of 52 cards. a) Determine your sets based on suite, face card and number cards. b) Determine the odds in favour and against drawing a face cardc) Determine the odds in favour and against drawing a red card d) Determine the odds in favour and against drawing a black number cardLesson 3 – Probability and OddsDetermining Odds from Probability1) Write the probability as a ratio2) Determine the probability in favour based off your original probability3) Write a second fraction for the probability that when added to the first would make one whole 4) Write this as a ratio to determine the odds againstExample 1: Research shows that the probability of an expectant mother, selected at random, at having twins is 132. Determine the odds for and against having twins. Example 2: 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 a male and the probability that it will be a female. Making Odds Equivalent 1) Multiply each set of odds to make the second term in the ratio equivalent . To do this you multiply the opposite odd by the second term in the oddExample 3: Make the following odds equivalent 15:28:5Example 4: A hockey game goes into a shootout. The coach has to decide which player should go in first for the shootout. The coach wants to use the best scorer first, so it will be based on the players shootout records. Player 1 has 13 attempts and 8 goals scored, and player two has 17 attempts and 10 goals scored. a) Write the odds in favour of each player scoring as a ratiob) Write the probability of each player as a percentc) Which player would you choose to shoot first? Which is easier to compare, odds or probability?Example 5: A group of Grade 12 students are holding a charity carnival to support a local animal shelter. The students have created a dice game that they call Bim and a card game that they call Zap. The odds against winning Bim are 5:2 and the odds against winning Zap are 7:3. Which game should you play if you love winning?Lesson 4 – Probabilities Using Counting MethodsDetermining Probabilities with Counting Methods 1) Determine the possible outcomes in favour. Make sure to take different conditions into consideration.2) Determine the total possible outcomes 3) Find the probability Example 1: Jamaal, Ethan and Alberto are competing with seven other boys to be on their schools cross-country team. All the boys a have an equal chance of winning the trial race. Determine the probability that Jamaal, Ethan, and Alberto will place first, second and third in any order. Example 2: A husband and wife meet a group of people and intrude their two daughters. They then explain how they had three other children at home. Determine the probability that at least one of the children is a boy. Example 3: You have 12 tiles that spell out the words SASKATCHEWAN. You place the tiles upside down and mix them up and rearrange them into a row. Determine the probability that when you flip the tiles back over spells the word Saskatchewan again.Example 4: There are 18 bikes in a spinning class. They are arranged in 3 rows, with 6 bikes in each row. 6 friends take the spin class, four of them are women and 2 of them are men, and want to be in the same row, but cannot request which bike they will use. Determine the probability that the six friends would be in the same row with the two men at either end. Lesson 5 – Mutually Exclusive EventsMutually Exclusive – Two or more events that cannot occur at the same time; for example the sun setting and the sun rising are mutually exclusive. Written: P(A∪B) = P(A) + P(B)Example 1: You are playing Jumanji and you need to roll a 5 or an 8 to get your friend out of the jungle. a) Fill out the outcome table for the sums.1753235157480b) What is the probability that you will get your friend out of the jungle on your next turn? Determine the odds for and against. 00b) What is the probability that you will get your friend out of the jungle on your next turn? Determine the odds for and against. 123456123456c) Is rolling a 5 or an 8 a mutually exclusive event? Explain. d) If you roll doubles you get another turn. What is the probability of getting to go another turn?e) Would the probability of rolling doubles or a sum of 5 or 8 be mutually exclusive? Explain.Non-Mutually Exclusive – Two events, A and B, that are intersecting sets.Principle of Inclusion and Exclusion – The number of elements in the union of two sets is the sum of each set , less the intersected elements. Written: nA∪B=nA+nB-n(A∩B)Probability of Non-mutually exclusive events 1) Determine the intersected area of the events 2) Use the Principle of Inclusion and Exclusion to determine the probability of the events happening 3) Determine the total outcomes and find the probability Example 2: Determine the probability, from Example 1, that you would roll a 5, 8 or doubles using the outcome table. Determine the odds for and against.Lesson 6 – Mutually Exclusive Events Part 2Example 1: A school newspaper published the results of a recent survey. 62% skip breakfast, 24% skip lunch and 22% eat both breakfast and luncha) Are skipping breakfast and skipping lunch mutually exclusive events? Explain. b) Determine the probability that a randomly selected student skips breakfast but not lunch. Determine the odds for and against. c) Determine the probability that a randomly selected student skips at least one breakfast or lunch. Determine the odds for and against. Example 2: Reid’s mother buys a new washer and dryer set for $2500 with a 1-year warranty. She can buy a 3-year extended warranty for $450. She researches the repair statistics for this set and finds the following data. Should she buy the warranty? Justify your decision. ApplianceP(repair within extended warranty period)Average Repair CostWasher22%$400Dryer13%$300Both3%$700Example 3: A car manufacturer keeps a database of cars available for sale. For model A, the database reports that 43% have heated leather seats, 36% have a sunroof and 49% have neither. Determine the probability of a model A car at a dealership having both heated leather seats and a sunroof. Lesson 7 – Conditional Probability Dependent Events – Events whose outcomes are affected by each other.Conditional Probability – The probability of an event occurring given that another event has already occurred. Conditional Probability Notation – P(B|A) is read: The probability that event B will occur, given that event A has already occurred.Multiple Event Notation – P(A and B) = P(A) ? P(B|A)Multiple Events1) Determine the number of events and outcomes 2) Determine the probability of each event 3) Multiply the events to determine the probability 4) If there are multiple conditions, add the probability togetherExample 1: A computer manufacturer knows that, in a box of 100 chips, 3 will be defective. You draw 2 chips at random from the box. Determine the probability that you will choose: 1 defective chip, 2 defective chips and no defective chips.Example 2: You need to choose a number between 1 and 40. The number is a multiple of 4. Determine the probability that the number is also a multiple of 6. Show using a formula and a Venn Diagram. Example 3: According to a survey, 91% of Canadians own a cellphone. Of these people, 42% have a smartphone. Determine, to the nearest percent, the probability that any Canadian you met during the month in which the survey was conducted would have a smartphone. Example 4: A team has a 60% chance of winning on calm days and a 70% chance of winning on windy days. Tomorrow, there is a 40% chance of high winds. What is the probability the team will win, if there is no ties allowed. Lesson 8 – Independent Events Example 1: You are playing a die and coin game. Each turn consists of rolling a regular die and tossing a coin. Points are awarded for rolling a 6 on the die and/or tossing heads with the coin: 1 point for either outcome3 points for both outcomes0 points for neither outcomesDetermine the probability of getting each of the points on your first turn and verify. Example 2: There 1000 tickets in a raffle for two prizes. After each draw the winning ticket is returned to the drum so it might be drawn again. You bought 5 tickets. Determine the probability, to the tenth of a percent, that he will win at least one prize. ................
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