SPIRE MATHS Stimulating, Practical, Interesting, Relevant, Enjoyable Maths For All

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1 Probability: scratch cards TYPE: OBJECTIVE(S): DESCRIPTION: OVERVIEW: EQUIPMENT: Main Use the vocabulary of probability when interpreting results of an experiment; appreciate that random processes are unpredictable. 3 screens. 1 and 2 show scratch cards to find probabilities of winning and losing. On 3 you have 2 cards to decide which is the better. Scratch card probability and considering options. A photocopy master is available. It contains 7 different scratch cards. Pupils have to decide how to place the winning symbols and work out a corresponding probability of a win and a loss. TYPE: OBJECTIVE(S): DESCRIPTION: OVERVIEW: EQUIPMENT: Plenary Use the vocabulary of probability when interpreting results of an experiment; appreciate that random processes are unpredictable. 3 screens. 1 asks you to explain scratch card probabilities. 2 shows an animation about 'average winnings'. 3 is vocabulary. Scratch card probability and considering options. None. Probability: scratch cards... 1 Main Whiteboard and Screen information... 2 Plenary Whiteboard and Screen information... 5 Scratch Cards... 8 Spire Maths interactive files available in a flash format at: Unfortunately they will not work on ipads or iphones. Page 1 of 8

2 Main Whiteboard and Screen information Screen 1: Scratch cards You are shown a rectangular scratch card and asked to scratch away one square. The winning symbol is a sign but many squares have partial winning symbols. You are told if you have won or lost. You click Next to see the complete card and are asked to work out the probability of winning and losing with the card. You can use the keypad to enter the answers or click Show to see them. You are then told the prize money and asked if you would buy the card. Less than half the squares will contain a winning symbol. Screen 2 is the same except only one winning symbol is on each card but the prize money is more. Key points: many pupils often believe that they are good guessers and that they can 'consistently beat the odds' - they tend to remember wins and forget losses; probabilities must be given in lowest terms; emphasis should be given to the sum of the probabilities of winning and losing (which must be 1) and that you can calculate one answer from the other; you may also wish to consider what might happen with any given card in the long run. Page 2 of 8

3 Screen 2: More scratch cards You are shown a rectangular scratch card and asked to scratch away one square. The winning symbol is a sign but many squares have partial winning symbols. You are told if you have won or lost. You click Next to see the complete card and are asked to work out the probability of winning and losing with the card. You can use the keypad to enter the answers or click Show to see them. You are then told the prize money and asked if you would buy the card. Only one square will contain a winning symbol. Screen 1 is the same except there will be at least one winning square and no more than half will win. The prize money per win is less. Key points: many pupils often believe that they are good guessers and that they can 'consistently beat the odds' - they tend to remember wins and forget losses; probabilities must be given in lowest terms; emphasis should be given to the sum of the probabilities of winning and losing (which must be 1) and that you can calculate one answer from the other; you may also wish to consider what might happen with any given card in the long run. Page 3 of 8

4 Screen 3: Pick a card You are shown two rectangular scratch cards of the same size where the winning symbols are all coloured red. You are given the prize money for each card and asked which one would have been the better one to buy. One of them will contain only one symbol and the other will contain one or more winning squares. You are asked which would have been the better one to buy. Key points: the lottery works on the assumption that more people will play it if there is a bigger prize (and you may see that this is the same for your class), so more pupils may opt for the card with the larger prize; in many, but not all cases, the cards and prizes have been designed so that the 'seller' should make a profit over time with the cards since the probability of a win x prize is always 1 or less - this may be something you wish to discuss with your pupils; you may wish this to form part of cross curricular work on the disadvantages of gambling. Page 4 of 8

5 Plenary Whiteboard and Screen information Screen 1: Scratch cards You are shown a rectangular scratch card and asked to scratch away one square. The winning symbol is a sign but many squares have partial winning symbols. You are told if you have won or lost. You click Next to see the complete card and the probabilities associated with winning and losing with the card (given in their simplest form). You are asked to explain how you would get these probabilities. You are then told the prize money for the card (cost of 1) and asked to explain what you think would happen in the long term. At all times the probability of winning times the amount you win comes to 1 or less. This could be part of a lesson on gambling, noting that this is not the usual sort of scratch card one buys, since most 'commercial' scratch cards don't usually have the potential for both a win and a loss (you often scratch off all the sections to reveal a winning or losing card). Key points: pupils should discuss the probabilities and notice that the probability of losing = 1 - probability of winning rather than counting the symbols; some pupils will find it difficult to consider the long term and believe that you can consistently 'beat' the system. Page 5 of 8

6 Screen 2: Comparing scratch cards You are shown a rectangular scratch card face up and told the cost and the amount that you would win on the card. You can then see an animation about what you could expect over one year. The animation shows that after one year your 'average winnings' is the prize x the probability x 52. You then subtract the cost of the tickets ( 52) and always end at a loss. This will reflect any scratch card system since the promoters have to aim for a profit. Key points: the notion of 'average winnings' is not an easy one and pupils need to consider that people will do better (and worse than this) but over a longer term this will give a good representation of what happens; pupils could discuss why in any real-life application, such as the lottery, how many big winners there are and consider where all this money comes from. Page 6 of 8

7 Screen 3: Vocabulary Vocabulary present: Biased, Certain, Chance, Doubt, Equally likely, Even chance, Event, Fair, Fifty-fifty chance, Good chance, Impossible, Interval, Likelihood, Likely, No chance, Outcome, Poor, Possible, Probability, Probable, Random, Risk, Sample, Sample space, Statistic, Theory, Uncertain, Unfair, Unlikely. Spire Maths interactive files available in a flash format at: Unfortunately they will not work on ipads or iphones. Page 7 of 8

8 Scratch Cards Here are some scratch card blanks. Place winning symbols on them and write down the probability of winning and losing for each card. What is a fair prize for a win on your card? = = Page 8 of 8