Odds and Evens

4308 /www/nrich/html/content/id/4308/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p>Here is a set of numbered balls used for a game:<br></br> </p> <div style="text-align: center;"> <mdo:image alt="Set of balls: 2, 3, 4, 5, 6" height="89" src="odds1.png" width="130"></mdo:image></div> <p>To play the game, the balls are mixed up and two balls are randomly picked out together. For example:</p> <div style="text-align: center;"><mdo:image alt="one ball numbered 4 and one ball numbered 5" height="31" src="OandE2.gif" width="76"></mdo:image></div> <div>The numbers on the balls are added together: $4 + 5 = 9$</div> <div style="font-weight: bold;"><br></br> If the total is even, you win. If the total is odd, you lose.<br></br> <br></br> How can you decide whether the game is fair?<br></br> <br></br> <span style="font-weight: 400;">Here are three more sets of balls:</span><br></br> <br></br> <div style="text-align: center;"> <mdo:image alt="Set B: 1,3,5,6,7 Set C: 2,3,4,5,6,8 Set D 1,3,4,5,7,9" height="112" src="odds2.png" width="445"></mdo:image></div> <br></br> <br></br> Which set would you choose to play with, to maximise your chances of winning?</div> <div><br></br> What proportion of the time would you expect to win each game?<br></br> <br></br> Test your predictions using the interactivity.</div> <br></br> <div><a href="/content/id/4308/MusicalChairs.swf">Full Screen Version</a></div> <p><mdo:flash height="400" width="550"><param name="allowfullscreen" value="true" ></param><param name="allowfullscreen" value="true" ></param><param name="movie" value="/content/id/4308/MusicalChairs.swf" ></param><param name="flashplayerversion" value="7" ></param><param name="height" value="400" ></param><param name="width" value="550" ></param></mdo:flash><br></br> <br></br> <span style="font-weight: bold;">Is it possible to produce a fair game?</span><br></br> Can you find a set of balls where the chance of getting an even total is the same as the chance of getting an odd total?<br></br> <br></br> Can you find more than one such set?<br></br> </p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">Jamie from Wilson's School used the interactivity to see which gave the fairest chance of winning:</span><br></br> Set C seems to be the fairest, because out of 100 tries, 50 won and 50 lost.<br></br> <br></br> <br></br> <span class="editorial">Alvin, also from Wilson's school, showed one way of calculating if a game is fair:</span><br></br> There are many possible ways to decide whether the game is fair or not. When you look at the set A, you can clearly see 3 odd balls and 2 evens.<br></br> Below is a simple way to find if the game is fair:<br></br> 2+3=5<br></br> 2+4=6<br></br> 2+5=7<br></br> 2+6=8<br></br> 3+4=7<br></br> 3+6=9<br></br> 4+5=9<br></br> 4+6=10<br></br> 5+6=11<br></br> We need to count the total of odds and the total of the evens. There are 6 odd totals and 4 evens. This can be expressed as a ratio 6:4 which can be cancelled to 3:2 It can be concluded that the game is unfair.<br></br> You can also use a <a href="/content/id/4308/Alvin.pdf">probability table</a> to represent it.<br></br> <br></br> <span class="editorial">Jacob (Wilson's School) realised that he didn't need to work out the answers to each addition, but just whether the answer was odd or even, which he represented with O and E.</span><br></br> <br></br> <span class="editorial">James (from Wilson's too!) also used a listing method to get the probability for set A as 0.4, and then compared the theoretical probability with the experimental probability:</span><br></br> When played 100 times the win/lose ratio is 0.42. It evens it self out after more games:<br></br> 200 times=0.385<br></br> 300 times=0.407<br></br> 400 times=0.385<br></br> 500 times=0.388<br></br> <br></br> <span class="editorial">Fred, Johannes, and Lok, all from St Barnabas School listed all the ways of making odd and even numbers for the four sets.</span><br></br> <br></br> <span class="editorial">Charlie and Shaun (both from Wilson's School) worked out the probabilities for all four sets. You can see Shaun's solution</span> <a href="/content/id/4308/shaun.pdf" class="editorial">here</a><span class="editorial">.</span><br></br> <br></br> <span class="editorial">Akeel (Wilson's School) explained what he would do to maximise the chance of winning:</span><br></br> To decide whether the game is fair you can find all the possible results and from that you can find out if the amount of times you can win is the same as the amount of times you can lose - if it is, the game is fair and if it isn't, the game isn't fair.<br></br> <br></br> In the case of set A, the game is not fair - the probability of you winning is $\frac{8}{20}$, which can be simplified to $\frac{4}{10}$ or 40%.<br></br> The probability of winning with B is $\frac{12}{20}$ and the probability of winning with D is $\frac{20}{30}$. The probability of winning with set C is $\frac{14}{30}$.<br></br> To maximise my chance of winning I would play with either set B or set D.<br></br> <br></br> <span class="editorial">Paul (Wilson's School) also identified that Set D gave the best chance of winning, and Tim (Wilson's School) suggested WHY set D gave the best chance of winning:</span><br></br> I would select set D to play with. This is because all of the numbers apart from the four are odd. This maximises your chance of winning because when you add two odd numbers together, you get an even and you are going to be adding more odd and odd numbers together than odd and even. I worked out that by playing set D, you have a 2 in 3 chance of winning. <br></br> <br></br> <span class="editorial">Hannah from Munich International School pointed out the following:</span><br></br> If you had only odd numbers, all the combinations of numbers would be even, meaning you would win every time! And if you only had even numbers, all the combinations of numbers would be even, meaning, again, you would win all the time! <br></br> <br></br> <span class="editorial">Tom and Hussein (Wilson's School) found a set that produced a fair game:</span><br></br> To produce a fair set of numbers from what we found out you would need 4 balls, one odd and three even, or the other way round, one even and three odd.<br></br> <br></br> <span class="editorial">Elliot (Wilson's School) also found a fair set:</span><br></br> A fair set can happen. If you have:<br></br> 1, 3, 5, 2<br></br> 2, 4, 6, 1<br></br> These are fair, as three results are odd, and three even. Any other groups with three of odd or even, then the one of even or odd are also fair.<br></br> <br></br> <span class="editorial">Philip, Tahmid and Jacques (Wilson's School) worked out the probabilities but also noticed something interesting along the way:</span><br></br> We've also noticed that the total number of possibilities for each set of balls can be worked out by this formula, in which $n$ = number of balls and $t$ = total possibilities:<br></br> $t = n(n-1)$<br></br> <br></br> <span class="editorial">Krystof from Uhelny Trh in Prague sent us</span> <a class="editorial" href="/content/id/4308/krystof.pdf">this</a> <span class="editorial">solution.</span><br></br> <span class="editorial">Finally, Chi from Raynes Park High School sent us</span> <a class="editorial" href="/content/id/4308/chisolution.pdf">this</a> <span class="editorial">solution.</span><br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=4308">This problem</a> offers an opportunity to explore and discuss two types of probability: experimental and theoretical. The simulation generates lots of experimental data quickly, freeing time to focus on predictions, analysis and justifications. Calculating the theoretical probabilities provides a motivation for using sample space diagrams or perhaps tree diagrams.</div> <div> </div> <div>The final question in the problem offers the opportunity for exploration of a rich context where collaborative working makes it possible to tackle an otherwise unmanageable task.</div> <br></br> <h3>Possible approach</h3> <div>You may wish to use the start of <a href="http://nrich.maths.org/7405&part=">What Numbers Can We Make?</a> as a preliminary activity to get students thinking about the effect of combining odd and even numbers.</div> <div> </div> <div><span style="font-style: italic;">The notes that follow are in two parts: the first part for teachers who wish to use the activity for a single lesson on probability and sample space diagrams or tree diagrams, and the second part for teachers who wish to follow this up with a collaborative task that leads to interesting and unexpected results.</span></div> <div> </div> <div>Start by showing how the game is played using Set A with the interactivity (or using numbered counters in a bag). Play the game no more than ten times, so that students have a feel for the game but don't have sufficient results to draw conclusions about the probabilities. Then ask them to decide whether they think the game is fair, and to do some maths to support their decision.</div> <div> </div> <div>While students are working, circulate and observe the methods being used:</div> <div> </div> <div> <mdo:image width="500" height="307" src="odds3.png" alt="different methods of recording"></mdo:image></div> <div> </div> <div>Bring the class together and choose individuals who used different methods to explain what they did to the class, recording what they did on the board. Perhaps choose those who used less sophisticated methods first. Emphasise the merits of a sample space method rather than a listing method, to prepare students for tackling examples with a large number of balls. Those who are confident with tree diagrams may prefer to continue using them.</div> <div> </div> <div>Use the interactivity to confirm that the experimental probability matches closely to the theoretical probability that students have calculated. There are opportunities here for rich discussion about how closely we expect an experimental probability to match the theory.</div> <div> </div> <div>Now show sets B, C and D, and ask them to <span style="font-weight: bold;">think on their own, without writing,</span> about which of the four sets they would choose to play with, to maximise their chances of winning. Once they have had a short time to reflect, ask them to discuss in pairs their choice, and to justify their decisions (again, without writing). There is often disagreement about which set offers the best chance of winning, so bring the class together to compare ideas before setting them the task of calculating the probabilities - discourage them from using inefficient listing methods.</div> <div> </div> <div>Once the probabilities have been calculated, use the interactivity again to confirm that the experimental probability is close to the calculated one.</div> <div> </div> <div>Now write up on the board a set E, which contains four large even numbers and two large odd numbers. Make them large enough that calculations would be offputting! Ask the class to work in pairs to calculate the probability of winning with set E, and give them a short time frame in which to do this. The intention is to alert students that the numbers themselves don't matter, but the numbers of odds and evens is the important point. Set E has the same structure as Set C, so we already know the chance of winning. Then the class can be introduced to this sort of sample space diagram where odds and evens are collected together:</div> <div> </div> <div> <mdo:image width="500" height="245" src="odds5.png" alt="sample space diagrams"></mdo:image></div> <div> </div> <div style="text-align: center;"> ........................................................</div> <div style="text-align: center;"> </div> <div style="text-align: left;">Point out that none of the sets looked at so far yields a fair game. "How could we go about finding out whether there are any sets that would give a fair game?"</div> <div style="text-align: left;">One way of organising the search is to draw up a table on the board showing different combinations of odds and evens:</div> <div style="text-align: left;"> </div> <div style="text-align: left;"> <mdo:image width="500" height="400" alt="board divided into different numbers of odds and evens" src="odds4.jpg"></mdo:image></div> <div style="text-align: left;"> </div> <div style="text-align: left;">Those already identified as not being fair games (sets A, B, C and D) can be crossed off. Then divide the class into groups working on different combinations and ask them to report back. Students could record combinations that have been checked on the board with a tick or a cross to show whether they are fair or not. If something has two ticks or two crosses, it could be accepted as being confirmed. When disagreements arise, ask other groups to resolve them.</div> <div style="text-align: left;"> </div> <div style="text-align: left;">There will be opportunities while the class are working to stop everyone and share students' insights that will make the job easier. For example:</div> <div style="text-align: left;"> </div> <div style="text-align: left; margin-left: 40px;">"None of the combinations with zero will work because..."</div> <div style="text-align: left; margin-left: 40px;">"If 3 odds and 2 evens won't work, 2 odds and 3 evens won't either, because..."</div> <div style="text-align: left; margin-left: 40px;">"You can't have the same number of evens and odds because..."</div> <div> </div> <div>Eventually, there will be a sea of crosses on the board and just a few combinations that work (four, if the class have gone up to 9 balls in total). Ask the class to stop and consider what the fair sets have in common. This may lead to some new conjectures about the total number of balls, so organise the class to test the conjecture out on the next obvious total.</div> <div> </div> <div>Once there is some confirmation about the total number of balls needed for fair games, conjectures can also be made about how these should be split into odds and evens. Students can be set to work to test examples with large numbers, using the simplified sample space method above. Draw attention to how valuable it is to work collaboratively as part of a mathematical community, and how difficult it would have been to have reached the same insights working alone.</div> <div> </div> <div>Although it is unlikely that many students will be able to prove their conjectures algebraically on their own, <a href="/content/id/4308/Odds%20and%20Evens.pdf">this proof</a> may be sufficiently accessible to be worth sharing with the class. There are a number of ways of using this resource: </div> <ul> <li>To be presented as an elegant way of proving the ideas the students have discovered</li> <li>As a proof presented on the board for students to recreate for themselves after it's been rubbed out</li> <li>To be printed out and distributed to students for them to make sense of, and for them to annotate so that they can talk through the proof, line by line, for someone who hadn't met it yet.</li> <li>As a 'proof sorting' exercise where the proof is cut into sections and mixed up for students to reassemble into the correct order</li> </ul> <h3>Key questions</h3> <div>How can you decide if a game is fair?</div> <div>What are the most efficient methods for recording possible combinations?</div> <div>How can we make this difficult task (of finding a fair game) more manageable?</div> <br></br> <h3>Possible extension</h3> <span style="font-weight: 400;">The problem</span> <a href="http://nrich.maths.org/919&part=" style="font-weight: 400;">In a Box</a> <span style="font-weight: 400;">offers another context for exploring exactly the same underlying mathematical structure, and could be used as a follow-up problem a few weeks after working on this one.</span> <h3>Possible support</h3> <div>The first parts of this problem should be accessible to most students, and can be used for focussing on the benefits of using sample space diagrams instead of listing combinations.</div> <div> </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You could start by using the interactivity - what happens when you pick out two balls from set $A$ $100$ times?<br></br> How many times did you get an even total? Were you expecting this?<br></br> <br></br> Work out all the possible combinations for set $A$, to see if you can make sense of what happened.<br></br> How will you make sure you don't miss any combinations out?<br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">Congratulations to Sam, James and Esther who sent in correct solutions to this problem.</span></p> <p><span class="editorial">Several others of you were really close, but made slight errors.</span></p> <p class="editorial">Here is Esther's solution:</p> Set D would give the greatest proportion of even numbers.<br></br> <br></br> The possibilities for this set are:<br></br> (1,3) (1,4) (1,5) (1,7) (1,9)<br></br> (3,4) (3,5) (3,7) (3,9)<br></br> (4,5) (4,7) (4,9)<br></br> (5,7) (5,9)<br></br> (7,9)<br></br> of which 10 out of 15 are even, a probability of 2/3 .<br></br> <br></br> When you work out A in this way, the probability is 2/5, B is 3/5 and C is 7/15.<br></br> <br></br> <p class="editorial">Esther correctly spotted that if you could select any numbers with at least one odd, the best numbers would be if they were all odd - two odds always add to make an even .</p> If assume that picking 3 and 5 is same as picking 5 and 3, then: A - 4 out of possible 10 are even: 3+5, 2+6, 2+4, 6+4 B - 6 out of possible 10 are even: 3+1, 3+5, 3+7, 1+5, 1+7, 5+7 C - 7 out of possible 15 are even: 8+2, 8+6, 8+4, 3+5, 2+6, 2+4, 4+6 D - 10 out of possible 15 are even: 7+9, 7+3, 7+1, 7+5, 3+9, 3+1, 3+5, 9+1, 9+5, 5+1 If all five balls or all six balls were odd, every combination would give an even total. <br></br></mdoxml> 2 3 0 0 1 1 0 Odds and Evens Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers? Probability Experimental probability Probability Theoretical probability Probability Tree diagrams Numbers and the Number System Odd and even numbers Information and Communications Technology Interactivities