A goalkeeper has a probability of of saving a penalty.

2. 1. A goalkeeper has a probability of of saving a penalty. How many penalties would you expect him to save out of 50 penalties taken? He would be expected to save 30 penalties.

3. 2. A batch of 1,600 items is examined. The probability that an item from this batch is defective is 0·04. How many items from this batch are defective? P(Defective) = 0·04 1600 items = n Expected = p × n = 0·04 × 1600 = 64 Expect 64 items to be defective.

4. 3. In an experiment, a standard six-sided die was rolled 72 times. The results are shown in the table. Which number on the die was obtained the expected number of times? Therefore, 3 was rolled the expected number of times.

5. 4. In a random survey of the voting intentions of a local electorate, the following results were obtained: (i) Calculate the probability that a randomly selected voter will vote for: (a) Politician A Total votes = 165 + 87 + 48 = 300

6. 4. In a random survey of the voting intentions of a local electorate, the following results were obtained: (i) Calculate the probability that a randomly selected voter will vote for: (b) Politician B

7. 4. In a random survey of the voting intentions of a local electorate, the following results were obtained: (i) Calculate the probability that a randomly selected voter will vote for: (c) Politician C

8. 4. In a random survey of the voting intentions of a local electorate, the following results were obtained: (ii) (a) Politician A? If there are 7,500 people in the electorate, how many votes would you expect to be for: 7,500 people = n Expected = p × n

9. 4. In a random survey of the voting intentions of a local electorate, the following results were obtained: (ii) (b) Politician B? If there are 7,500 people in the electorate, how many votes would you expect to be for: 7,500 people = n Expected = p × n

10. 4. In a random survey of the voting intentions of a local electorate, the following results were obtained: (ii) (c) Politician C? If there are 7,500 people in the electorate, how many votes would you expect to be for: 7,500 people = n Expected = p × n

11. 5. In a raffle, 250 tickets are sold at €1 each for three prizes of €100, €50 and €10. You buy one ticket. (i) What is the expected value of this raffle? Expected value: Σx.p(x)

12. 5. In a raffle, 250 tickets are sold at €1 each for three prizes of €100, €50 and €10. You buy one ticket. (ii) Does it represent a good investment of €1? Explain your answer. It does not represent a good investment of €1 as the expected value is 0·64 which means if a large number of people bought tickets the average win is below the pay in amount. You would expect to lose 0·36. Net “winnings” = 0·64 – 1 = – 0·36

13. 6. Jennifer is playing a game at an amusement park. There is a 0·1 probability that she will score 10 points, a 0·2 probability that she will score 20 points, and a 0·7 probability that she will score 30 points. How many points can Jennifer expect to receive by playing the game? Expected value = Σx.p(x) Σx.p(x) = 1 + 4 + 21 = 26 points

14. 7. You take out a fire insurance policy on your home. The annual premium is €300. In case of fire, the insurance company will pay you €200,000. The probability of a house fire in your area is 0·0002. What is the expected value? (i) Expected value = Σx·p(x) = 40 + 0 = 40

15. 7. You take out a fire insurance policy on your home. The annual premium is €300. On case of fire, the insurance company will pay you €200,000. The probability of a house fire in your area is 0·0002. (ii) What is the insurance company’s expected value? The expected value is the same for the insurance company and customer. However, the perspective is changed. The customer is down €260 since they paid €300 for the policy, the company is up €260.

16. 7. You take out a fire insurance policy on your home. The annual premium is €300. On case of fire, the insurance company will pay you €200,000. The probability of a house fire in your area is 0·0002. (iii) Suppose the insurance company sells 100,000 of these policies. What can the company expect to earn? Profit on 1 policy: = €260 100,000 policies : 100,000 × €260 = €26,000,000

17. 8. A hundred tickets are sold for a movie at the cost of €10 each. Some tickets have cash prizes as a part of a promotional campaign: one prize of €50, three prizes of €25 and five prizes of €20. What is the expected value if you buy one ticket? Expected value: Σx.p(x)

18. 9. You pay €10 to play the following game of chance. There is a bag containing 12 balls, five are red, three are green and the rest are yellow. You are to draw one ball from the bag. You will win €14 if you draw a red ball and you will win €12 if you draw a green ball. How much do you expect to win or lose if you play this game 100 times?

19. 9. You pay €10 to play the following game of chance. There is a bag containing 12 balls, five are red, three are green and the rest are yellow. You are to draw one ball from the bag. You will win €14 if you draw a red ball and you will win €12 if you draw a green ball. How much do you expect to win or lose if you play this game 100 times? Pay €10 – Expected value €8·83 so “nett winnings” = 8·83 – 10 =

20. 10. A sports club decides to hold a field day to raise money for local charities. One of the games involves spinning the wheel shown on the right. You win the amount the pointer lands on. It costs €5 to play the game. (i) What is the expected value of the game? Expected value = Σx·p(x) €4·63 is the expected value of the game.

21. 10. A sports club decides to hold a field day to raise money for local charities. One of the games involves spinning the wheel shown on the right. You win the amount the pointer lands on. It costs €5 to play the game. (ii) How could they adjust the wheel to ensure they make more money for the charities? In order to make more money for the charities they could lower the amount that is won in each or some segment(s). Or They could enlarge the €2 segment and make the €5 segment smaller.

22. 11. (i) Which offer has the greater expected value? A child asks his parents for some money. The parents make the following offers. Father’s offer: The child flips a coin. If the coin lands heads-up, the father will give the child €20. If the coin lands tails-up, the father will give the child nothing. Mother’s offer: The child rolls a 6-sided die. The mother will give the child €3 for each dot on the up side of the die. Expected value = Σx.p(x) Father’s offer Σx.p(x) = €10

23. 11. (i) Which offer has the greater expected value? A child asks his parents for some money. The parents make the following offers. Father’s offer: The child flips a coin. If the coin lands heads-up, the father will give the child €20. If the coin lands tails-up, the father will give the child nothing. Mother’s offer: The child rolls a 6-sided die. The mother will give the child €3 for each dot on the up side of the die. Mother’s offer

24. 11. (i) Which offer has the greater expected value? A child asks his parents for some money. The parents make the following offers. Father’s offer: The child flips a coin. If the coin lands heads-up, the father will give the child €20. If the coin lands tails-up, the father will give the child nothing. Mother’s offer: The child rolls a 6-sided die. The mother will give the child €3 for each dot on the up side of the die. Mother’s offer €10·50 > €10 Therefore, the mother’s offer is greater.

25. 11. (ii) Which offer would you choose if you were the child? Justify your answer. A child asks his parents for some money. The parents make the following offers. Father’s offer: The child flips a coin. If the coin lands heads-up, the father will give the child €20. If the coin lands tails-up, the father will give the child nothing. Mother’s offer: The child rolls a 6-sided die. The mother will give the child €3 for each dot on the up side of the die. I would choose the mother’s offer as over many times the expected outcome would be higher.

26. 12. (i) What is the expected value for each of the following investment packages for a €1,000 investment? Conservative investment Speculative investment • Complete loss: 40% chance • Complete loss: 1% chance • No gain or loss: 15% chance • No gain or loss: 35% chance • 100% gain: 15% chance • 10% gain: 59% chance • 400% gain: 15% chance • 20% gain: 5% chance • 900% gain: 15% chance Expected value = Σx.p(x) €1,000 investment

27. 12. (i) What is the expected value for each of the following investment packages for a €1,000 investment? Speculative: Σx.p(x) = −400 + 0 + 150 + 450 + 1,200 = 1,400

28. 12. (i) What is the expected value for each of the following investment packages for a €1,000 investment? Speculative: Expected payout = €1,400 on a €1,000 investment Expected value = €2,400

29. 12. (i) What is the expected value for each of the following investment packages for a €1,000 investment? Conservative: Σx.p(x) = –10 + 0 + 59 + 10 = 59

30. 12. (i) What is the expected value for each of the following investment packages for a €1,000 investment? Conservative: Expected payout = €59 on a €1,000 investment Expected value = €1,059

31. 12. (ii) Which would you choose? Why? There are two possible answers here: I would choose: The speculative investment because the expected value is higher so it should earn more. Or The conservative because the risk of losing everything is lower.

32. 13. A biased die is used in a fairground game. The probabilities of getting the six different numbers on the die are shown in the table below. (i) Find the expected value of the random variable x, where x is the number thrown. Expected value = Σx.p(x) Σx.p(x) = 0·12 + 0·52 + 0·3 + 0·92 + 0·9 + 0·66 = 3·42 Expected value = €3·42

33. 13. A biased die is used in a fairground game. The probabilities of getting the six different numbers on the die are shown in the table below. (ii) It costs €3·50 to play the game. The player rolls a die once and wins the number of euro shown on the die. By doing the calculations required, complete the following sentence: ‘If you play the game many times with a fair die, you will win an average of ______ per game, but if you play with the biased die you will lose an average of ______ per game.’

34. 13. A biased die is used in a fairground game. The probabilities of getting the six different numbers on the die are shown in the table below. (ii) Playing with a fair die Expected value = €3·50

35. 13. A biased die is used in a fairground game. The probabilities of getting the six different numbers on the die are shown in the table below. (ii) Biased die net winnings Fair die net winnings = expected value – cost Expected value – cost = 3·50 – 3·50 = 3·42 – 3·50 = 0 = –0·08 If you play the game many times with a fair die, you will win an average of €0·00 per game, but if you play with the biased die you will lose an average of €0·08 per game.