Probability of Events in Games

2. 1. Jamie is playing a game where he must toss a coin until he gets a head. What is the probability that he gets a head for the first time on: the first throw? (i) Success = Flipping a head P (Failure) = P (Success) = P (Head on first flip) =

3. 1. Jamie is playing a game where he must toss a coin until he gets a head. What is the probability that he gets a head for the first time on: the second throw? (ii)

4. 1. Jamie is playing a game where he must toss a coin until he gets a head. What is the probability that he gets a head for the first time on: the third throw? (iii)

5. 1. Jamie is playing a game where he must toss a coin until he gets a head. What is the probability that he gets a head for the first time on: his fifth throw? (iv)

6. 2. Ciaran is playing a game using a fair die. He must throw the die until he gets a 6. What is the probability that he throws a 6 in: Success = throwing a 6

7. 2. Ciaran is playing a game using a fair die. He must throw the die until he gets a 6. What is the probability that he throws a 6 in: one throw? (i) Success = throwing a 6

8. 2. Ciaran is playing a game using a fair die. He must throw the die until he gets a 6. What is the probability that he throws a 6 in: two throws? (ii) Success = throwing a 6

9. 2. Ciaran is playing a game using a fair die. He must throw the die until he gets a 6. What is the probability that he throws a 6 in: three throws? (iii) Success = throwing a 6

10. 2. Ciaran is playing a game using a fair die. He must throw the die until he gets a 6. What is the probability that he throws a 6 in: his tenth throw? (iv) Success = throwing a 6 P(Success on 10th throw)

11. 3. A fair die is thrown three times. What is the probability that a 2 or a 3 is thrown: Success = 2 or 3

12. 3. A fair die is thrown three times. What is the probability that a 2 or a 3 is thrown: for the first time on the third throw? (i)

13. 3. A fair die is thrown three times. What is the probability that a 2 or a 3 is thrown: once in three throws? (ii) P (FFS) or P (SFF) or P (FSF)

14. 4. A spinner has five equal sections. Three of them are green and two of them are yellow. Chloe is playing a game where she has to spin the spinner until it lands on yellow. What is the probability that she lands on yellow for the first time on:

15. 4. A spinner has five equal sections. Three of them are green and two of them are yellow. Chloe is playing a game where she has to spin the spinner until it lands on yellow. What is the probability that she lands on yellow for the first time on: the first spin? (i) P(Yellow) = P(S) =

16. 4. A spinner has five equal sections. Three of them are green and two of them are yellow. Chloe is playing a game where she has to spin the spinner until it lands on yellow. What is the probability that she lands on yellow for the first time on: the third spin? (ii) P(GGY) = P(FFS)

17. 4. A spinner has five equal sections. Three of them are green and two of them are yellow. Chloe is playing a game where she has to spin the spinner until it lands on yellow. What is the probability that she lands on yellow for the first time on: the sixth spin? (iii)

18. 4. A spinner has five equal sections. Three of them are green and two of them are yellow. Chloe is playing a game where she has to spin the spinner until it lands on yellow. What is the probability that she lands on yellow for the first time on: the ninth spin? (iv) Give your answer in scientific notation. P(F)8 × P(S)

19. 5. Zoe is the best free-throw taker on her basketball team. Her average probability of scoring from a free shot is 0·78. If she takes three free throws in a game, what is the probability that she will: P(F) = 1 – P(S) P(S) = 0·78 = 1 – 0·78 = 0·22

20. 5. Zoe is the best free-throw taker on her basketball team. Her average probability of scoring from a free shot is 0·78. If she takes three free throws in a game, what is the probability that she will: score all three? (i) P(SSS) = 0·78 × 0·78 × 0·78 = 0·474552

21. 5. Zoe is the best free-throw taker on her basketball team. Her average probability of scoring from a free shot is 0·78. If she takes three free throws in a game, what is the probability that she will: score two out of three? (ii) P (Scores 2 out of 3) P(SSF) or P(SFS) or P(FSS) = (0·78 × 0·78 × 0·22) + (0·78 × 0·22 × 0·78) + (0·22 × 0·78 × 0·78) = 0·401544

22. 5. Zoe is the best free-throw taker on her basketball team. Her average probability of scoring from a free shot is 0·78. If she takes three free throws in a game, what is the probability that she will: score only on the final throw? (iii) P (Score on final throw) P(FFS) = (0·22 × 0·22 × 0·78) = 0·037752

23. 5. Zoe is the best free-throw taker on her basketball team. Her average probability of scoring from a free shot is 0·78. If she takes three free throws in a game, what is the probability that she will: not score at all? (iv) P (No score) P(FFF) = 0·22 × 0·22 × 0·22 = 0·010648

24. 5. Zoe is the best free-throw taker on her basketball team. Her average probability of scoring from a free shot is 0·78. If she takes three free throws in a game, what is the probability that she will: score at least once? (v) P (Scores at least once) = P (Scores once or twice or three times) = P(SFF) or P(FSF) or P(FFS) or P(SSF) or P(SFS) or P(FSS) or P(SSS) = (0·22 × 0·22 × 0·78) + (0·78 × 0·22 × 0·22) + (0·22 × 0·78 × 0·22) + 0·401544 + 0·474552 = 0·037752 + 0·037752 + 0·037752 + 0·401544 + 0·474552 = 0·989352

25. 6. At the end of Maths class, a teacher gives her students a three-question multiple-choice test to check if they have understood the lesson. Each question has three options. A student guesses on all questions. What is the probability that he gets: Success = Guessing the correct answer out of the 3 options 3 options

26. 6. At the end of Maths class, a teacher gives her students a three-question multiple-choice test to check if they have understood the lesson. Each question has three options. A student guesses on all questions. What is the probability that he gets: all questions correct? (i)

27. 6. At the end of Maths class, a teacher gives her students a three-question multiple-choice test to check if they have understood the lesson. Each question has three options. A student guesses on all questions. What is the probability that he gets: two questions correct? (ii)

28. 6. At the end of Maths class, a teacher gives her students a three-question multiple-choice test to check if they have understood the lesson. Each question has three options. A student guesses on all questions. What is the probability that he gets: no questions correct? (iii)

29. 6. At the end of Maths class, a teacher gives her students a three-question multiple-choice test to check if they have understood the lesson. Each question has three options. A student guesses on all questions. What is the probability that he gets: the first two correct and the final question incorrect? (iv)

30. 7. In a survey of her class, Olivia found that 35% of the students walk to school. Three students are selected at random from the class. Giving your answers as a decimal, what is the probability that: 35% walk Success = A student walks P(S) = 0·35 P(F) = 1 – 0·35 = 0·65

31. 7. In a survey of her class, Olivia found that 35% of the students walk to school. Three students are selected at random from the class. Giving your answers as a decimal, what is the probability that: all three students walk to school? (i) P(SSS) = 0·35 × 0·35 × 0·35 = 0·042875

32. 7. In a survey of her class, Olivia found that 35% of the students walk to school. Three students are selected at random from the class. Giving your answers as a decimal, what is the probability that: none of the three students walk to school? (ii) P(FFF) = 0·65 × 0·65 × 0·65 = 0·274625

33. 7. In a survey of her class, Olivia found that 35% of the students walk to school. Three students are selected at random from the class. Giving your answers as a decimal, what is the probability that: two out of the three students walk to school? (iii) P(SSF) or P(SFS) or P(FSS) = (0·35 × 0·35 × 0·65) + (0·35 × 0·65 × 0·35) + (0·65 × 0·35 × 0·35) = 0·079625 + 0·079625 + 0·079625 = 0·238875

34. 8. Charlie is the goalkeeper on the school soccer team. His average success rate for saving goals is If there are three scoring opportunities in a match, what is the probability that:

35. 8. Charlie is the goalkeeper on the school soccer team. His average success rate for saving goals is If there are three scoring opportunities in a match, what is the probability that: he saves all three? (i)

36. 8. Charlie is the goalkeeper on the school soccer team. His average success rate for saving goals is If there are three scoring opportunities in a match, what is the probability that: he doesn’t save any of them? (ii)

37. 8. Charlie is the goalkeeper on the school soccer team. His average success rate for saving goals is If there are three scoring opportunities in a match, what is the probability that: he saves the first two but not the third? (iii)

38. 8. Charlie is the goalkeeper on the school soccer team. His average success rate for saving goals is If there are three scoring opportunities in a match, what is the probability that: he saves one? (iv) P (one success) = P(SFF) or P(FSF) or P(FFS)

39. Stephen comes to school by car. On his way to school he passes three sets of traffic lights. The probability that the lights will be red is . What is the probability that: 9.

40. Stephen comes to school by car. On his way to school he passes three sets of traffic lights. The probability that the lights will be red is . What is the probability that: 9. the first set of lights will be green? (i)

41. Stephen comes to school by car. On his way to school he passes three sets of traffic lights. The probability that the lights will be red is . What is the probability that: 9. the first time the lights are red is at the last set of lights? (ii)

42. Stephen comes to school by car. On his way to school he passes three sets of traffic lights. The probability that the lights will be red is . What is the probability that: 9. all three sets of lights are red? (iii)

43. Stephen comes to school by car. On his way to school he passes three sets of traffic lights. The probability that the lights will be red is . What is the probability that: 9. at least one set of lights are red? (iv) P (At least one S) = P (FFS) or P (FSF) or P (SFF) or P (FSS) or P (SFS) or P (SSF) or P (SSS)

44. 10. Suppose a student takes a multiple choice test. The test has 10 questions, each of which has four possible answers (only one correct). (i) If the student guesses the answer to each question, do the questions form a sequence of Bernoulli trials? Explain your answer. Yes. Each question is independent and has two options; success or failure. Therefore, the test forms a series of Bernoulli trials.

45. 10. Suppose a student takes a multiple choice test. The test has 10 questions, each of which has four possible answers (only one correct). (ii) List the possible outcomes. Possible outcomes: 1. Success – the guessis correct 2. Failure – the guess is incorrect

46. 10. Suppose a student takes a multiple choice test. The test has 10 questions, each of which has four possible answers (only one correct). (iii) Write the probability associated with each outcome.