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Probability Revision WorksheetLeaving Cert - Ordinary LevelSyllabusYou should be able to:Explain / understand the following key wordsSample spaceSystematic listingTwo – way tableTree diagramFundamental principle of countingTrial / OutcomeSet (sample space)EventDesirable outcomesBiased / fair dieLikelihood scale / Probability scaleRelative frequency / Expected frequencyIndependent eventsMutually exclusive eventsExpected valueBernoulli TrialsArrangementsCalculate probability List all possible outcomes using; Systematic listing / Two – way tables / Tree diagrams Use the Fundamental Principle of Counting to list all possible outcomes Calculate relative frequency and fairness Calculate expected frequency Calculate expected value and recognize its role in decision makingSolving probability questions using counting methods Count the number of ways of arranging n distinct objectsCalculating the probability of independent events / mutually exclusive eventsSolving problems involving Bernoulli trials (what is a Bernoulli trial?)Question 1: Calculating ProbabilityA fair dice is rolled, what is the probability of getting - an odd number- a number less than 3- a prime numberA card is drawn from a pack of 52 playing cards. What is the probability that the card is- a red card- a spade- a king- a red 10A bag contains 4 red beads, 3 black beads and 7 green beads. If a bead is picked at random from the bag, find the probability that the bead is- red- not black- red or greenFrom the word PROBABILITY, what it the probability of - choosing the letter B- choosing a vowel- choosing an A or I*Exam QuestionPeter and Niamh go to a large school. One morning, they arrive early. While they are waiting, they decide to guess whether each of the next three students to come in the door will be a boy or a girl.(i) Write out the sample space showing all the possible outcomes. For example, BGG is one outcome, representing Boy, Girl, Girl.(ii) Peter says these outcomes are equally likely. Niamh says they are not. What do you need toknow about the students in the school to decide which of them is correct?(iii) If all the outcomes are equally likely, what is the probability that the three students will be two girls followed by a boy?(iv) Niamh guesses that there will be at least one girl among the next three students. Peter guesses that the next three students will be either three boys or two boys and a girl. Who is more likely to be correct, assuming all outcomes are equally likely? Justify your answer.* Exam Question(i) Two fair coins are tossed. What is the probability of getting two heads?(ii) Two fair coins are tossed 1000 times. How often would you expect to get two heads?(b) Síle hands Pádraig a fair coin and tells him to toss it ten times. She says that if he gets ten heads she will give him a prize. The first nine tosses are all heads. How likely is it that the last toss will also be a head? Tick the correct answer, and give a reason.Extremely unlikely …Fairly unlikely …50-50 chance …Fairly likely …Almost certain …Reason:Question 2: Two-Way TablesIf two dice are thrown and the scores are added, draw a two-way table to show all possible outcomes. Hence find the probability that - the total is exactly 7- the total is 11 or more- the total is a multiple of 5- the total is 6 or 8 Two coins are tossed. Draw a two-way table to represent the data. Hence find the probability that- Two heads were thrown- One head and one tail was throwQuestion 3: Relative & Expected FrequencyJohn threw a coin 200 times. If the coin landed on tails 115 times how many times would you expect it to land on tails if the coin was flipped 650 times. A dice was rolled 35 times and it landed on the number 1 four times. Using these results, how many times would you expect to land on the number 1 out of 175 rolls?ColourRedBlueGreen YellowPurpleProbability0.350.10.150.05 The probability of the spinner landing on each of the 5 colours are given below. Use the results to answer the following.i) What is the probability of the spinner landing on purple?ii) What colour is the spinner most likely to land on?iii) If the spinner is spun 200 times, how many times would you expect to land on green?* Exam QuestionA plastic toy is in the shape of a hemisphere. When it falls on the ground, there are two possible outcomes: it can land with the flat side facing down or with the flat side facing up. Two groups of students are trying to find the probability that it will land with the flat side down. (a) Explain why, even though there are two outcomes, the answer is not necessarily equal to ?. (b) The students estimate the probability by experiment. Group A drops the toy 100 times. From this, they estimate that it lands flat side down with probability 0·76. Group B drops the toy 500 times. From this, they estimate that it lands flat side down with probability 0·812. - Which group’s estimate is likely to be better, and why? - How many times did the toy land flat side down for Group B? - Using the data from the two groups, what is the best estimate of the probability that the toy lands flat side down?Question 4: Mutually Exclusive & Non-Mutually Exclusive EventsA dice is thrown. Find the probability that the number showing is- a 5- an even number- a 5 or an even numberA dice is thrown. Find the probability that the number showing is- a 2- an odd number- a 2 or an odd numberA card is selected a random from a pack. What is the probability that it is- a diamond- a black picture card- a diamond or a black picture cardA card is selected a random from a pack. What is the probability that it is- a heart- a queen- a heart or a queenA card is selected a random from a pack. What is the probability that it is- a red card- a 10- a red card or a 10A dice is thrown. What is the probability of getting- an even number- a prime number- an even number or a prime numberA pair of dice are thrown. What is the probability of getting- a total of 12- same number on both dice- a total of 12 and the same number on both dice* Exam Question(A) A garage has 5 black cars, 9 red cars and 10 silver cars for sale. A car is selected at random. What is the probability that:- The car is black?- The car is black or red?(B) A car is selected at random. Then a second car is selected at random from those remaining. What is the probability that:- The first car is silver and the second car is black?- One of the selected cars is red and the other is black?(C) Three of the black cars, two of the red cars and four of the silver cars have diesel engines. One car from the garage is again selected at random. What is the probability that - it is a red car or a diesel car?Question 5: Venn Diagrams & Probability125730087820500*Exam QuestionIn the Venn diagram below, the universal set is a normal deck of 52 playing cards. The two sets shown represent clubs and picture cards (kings, queens and jacks).(a) Show on the diagram the number of elements in each region.(i) A card is drawn from a pack of 52 cards. Find the probability that the card drawn is the king of clubs.(ii) A card is drawn from a pack of 52 cards. Find the probability that the card drawn is a club or a picture card.(iii) Two cards are drawn from a pack of 52 cards. Find the probability that neither of them is a club or a picture card. Give your answer correct to two decimal places.Question 6: Multiplication Rule & Bernoulli TrialsA coin is tossed and a dice is thrown. What is the probability of getting- a tail and an even number- a head and a 2- a tail and a factor of 6Two dice are thrown. What is the probability of getting- 2 threes- 2 odd numbers- both dice showing a number greater than 4The letters of the word ALGEBRA are written on individual cards and placed into a bag. A card is selected at random and the replaced. A second card is then selected. Find the probability of obtaining- the letter B and then the letter G in that order- the letter A twice- the letter B and A (and order)John and Bob celebrate their birthdays in a particular week. Assuming the birthdays are equally likely to fall on any day of the week, calculate the probability that- John’s birthday is at the weekend- both birthdays are on a Tuesday- both birthdays are during the weekWhat is a Bernoulli trial?Jenna tosses a fair coin several times. Find the probability for each of the following:- the first head to occur on the second toss- the first head to occur on the fifth tossKaren throws a fair dice until she gets a 2. Calculate the probability that she gets- a 2 on the first throw- the first 2 on the third throw- the first 2 on the fifth throwThe probability that it will rain on a given day in January is 0.6. Find the probability that - the first day has no rain- the third day is the first day to have rain- the fourth day is the first day to have rainJohnny plays a series of snooker matches against the same opponent. The probability that he wins any game is ?. Calculate the probability that- Johnny has his first win in his second game.- He has his first win in the third game.- He loses all three games.* Exam QuestionWhen taking a penalty kick, the probability that Kevin scores is always ?.(a) Kevin takes a penalty. What is the probability that he does not score?(b) Kevin takes two penalties. What is the probability that he scores both?(c) Kevin takes three penalties. What is the probability that he scores exactly twice?(d) Kevin takes five penalties. What is the probability that he scores for the first time on his fifth penalty?Question 7: Tree DiagramsA fair coin is tossed two times. Draw a tree diagram to show all possible outcomes. Work out the probability of getting- two heads- one head and one tail (any order)Box A contains 3 red beads and 4 blue beads. Box B contains 2 red beads and 3 blue beads.One bead is taken at random from each box.- Draw a tree diagram to show all of the outcomes- Hence calculate the probability that two red beads are drawn.- Hence calculate the probability that one red and one blue bead is drawnA fair coin is tossed three times. Draw a tree diagram to show all possible outcomes. Work out the probability of getting- three heads- two heads and one tail (any order)Bag A contains 3 green beads and 2 white beads. Bag B contains 4 green beads and 1 white bead.One bead is taken at random from each bag.- Draw a tree diagram to show all of the outcomes- Hence calculate the probability that two white beads are drawn.- Hence calculate the probability that one green and one white bead is drawnQuestion 8: Expected ValueIn a casino, it costs €3 to throw a dice. If you roll a 1 you win €1, a 2 you win €2, a 3 you win €3,etc… Calculate how much you are expected to win or lose if you play this game and hence is it worth playing the game?43434002032000Find the expected value when this spinner is spun a large number of times. If it costs €7 to spin the wheel, is it worth playing?In a casino, a game consists of throwing a pair of dice and adding the scores. If you score a total of 7, you win €24. If you score a total of 9, you lose €27. For all other scores you neither win nor lose. If you play this game, what do you expect to win or lose?Player Wins€0€1€2€3Required Outcomes*Exam QuestionAn unbiased circular spinner has a movable pointer and five equal sectors, two coloured green and three coloured red.(a) (i) Find the probability that the pointer stops on green for one spin of the spinner.(ii) List all the possible outcomes of 3 successive spins of the spinner. (b) A game consists of spinning the spinner 3 times. Each time the spinner stops on green the player wins €1; otherwise the player wins nothing. For example, if the outcome of one game is “green, red, green” the player wins €plete the following table: Is one spin of the spinner above an example of a Bernoulli trial?Answer: ___________________Explain what a Bernoulli trial is.Question 9: Fundamental Principle of CountingState the fundamental principle of counting.A dice is rolled and a coin is tossed. How many outcomes are possible?List all possible outcomes.A menu is offering 3 choices of starter, 4 main course options and 2 types of deserts. How many different three-course meals are possible?An iPhone code consists of a 4 digit codes using numbers from 0 to 9. How many codes are possible?There are 4 roads from A to B and 5 roads from B to C. How many ways can you get from A to C?* Exam QuestionHelen has enough credit to download three songs from the internet. There are seven songs that she wants.(i) How many different possible selections of three songs can she make?(ii) If there is one particular song that she definitely wants, how many different selections can she now make?Question 10: ArrangementsIn how many ways can the letters of the word ALGEBRA be arranged?How many of these arrangements begin with a vowel?How many of these arrangements begin with an L and end with an R?In how many ways can the letters of the word ORANGE be arranged?How many of these arrangements begin with a vowel and end with a vowel?How many of these arrangements begin with an N and end with an R?How many of these arrangements have letter NG together?In how many ways can the letters of the word MATHEMATICS be arranged?How many of these arrangements begin with a M and end with a T?How many of these arrangements begin with a vowel?How many of these arrangements have letter CS together?Eight horses run a race. Assuming that all horses finish the race, how many ways can the horses finish the race?In how many ways can the first three places be filled?*Exam QuestionA bank issues a unique six-digit password to each of its online customers. The password may contain any of the numbers 0 to 9 in any position and numbers may be repeated. For example, the following is a valid password: 071737(a) How many different passwords are possible? (b) How many different passwords do not contain any zero? (c) One password is selected at random from all the possible passwords. What is the probability that this password contains at least one zero? (d) John is issued with one such password from the bank. Each time John wants to access his account online, the bank’s website requires him to input three of his password digits into the boxes provided. For example, he may be asked for the 2nd, 4th and 5th digits, as shown below. * __ * __ __ *In how many different ways can the bank select the three required boxes? ................
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