2857 /www/nrich/html/content/id/2857/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Two children are playing a game with three balls, two blue ones and one red one. <br></br> Terry is red and Ali is blue.<br></br> <br></br> <div style="text-align: center;"><mdo:image width="110" height="95" alt="two blue balls and a red ball" src="2and1.jpg"></mdo:image></div> <br></br> They toss up the balls, which run down a slope so that they land in a row of three, like this:<br></br> <br></br> <div style="text-align: center;"> <mdo:image width="160" height="50" src="2and1A.jpg" alt="blue, red, blue balls in a line"></mdo:image></div> <br></br>The colour in the middle wins that round. Is this a fair game?<br></br> Who do you think will win? <br></br> Why? <br></br> <br></br> Try using the interactivity below to get a feel for the game. (The sum column tells you how many times Terry has won so far.) <br></br> <br></br> <a href="/content/id/2857/MusicalChairs.swf"> Full Screen Version</a> <br></br> <mdo:flash height="400" width="550"><param name="movie" value="/content/id/2857/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></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">We had some good solutions to this problem. Rukmini from Dowanhill Primary wrote:</p> <div>I first got some red and blue wooden pieces. Next, I sorted out these ones.<br></br> <div style="clear: both;">The combinations were:<br></br> <div style="clear: both;">Blue Red Blue (Terry wins)<br></br> <div style="clear: both;">Red Blue Blue (Ali wins)<br></br> <div style="clear: both;">Blue Blue Red (Ali wins)<br></br> <div style="clear: both;">No, this is not a fair game because Terry only has one winning set and Ali has two possible winning sets. It's more likely that Ali will win.</div> <div style="clear: both;"></div> <p style="clear: both;"><span class="editorial">You're right, Rukmini, well done. Nicholas who goes to Lochinver House School and Courtney from Holystone Primary expressed their solution in a slightly different, but equally as good, way. Courtney said:</span></p> <p style="clear: both;">It is not fair because Terry has a one in three chance but Ali has a $2$ in $3$ chance.</p> </div> </div> </div> </div> </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Two and One</h2> <br></br> Two children are playing a game with three balls, two blue ones and one red one.<br></br> Terry is red and Ali is blue.<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="two blue balls and a red ball" height="95" src="2and1.jpg" width="110"></mdo:image></div> <br></br> They toss up the balls, which run down a slope so that they land in a row of three, like this:<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="blue, red, blue balls in a line" height="50" src="2and1A.jpg" width="160"></mdo:image></div> <br></br> The colour in the middle wins that round. Is this a fair game?<br></br> Who do you think will win?<br></br> Why?<br></br> <br></br> Try using the interactivity below to get a feel for the game. (The sum column tells you how many times Terry has won so far.)<br></br> <br></br> <a href="/content/id/2857/MusicalChairs.swf">Full Screen Version</a><br></br> <mdo:flash height="400" id="/content/id/2857/MusicalChairs.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2857/MusicalChairs.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash></div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>The interactivity will help children to get a feel for <a href="http://nrich.maths.org/public/viewer.php?obj_id=2857&amp;part=index">this problem</a>, however they should be encouraged not just to identify a pattern in the data they collect, but to explain why this pattern occurs.</div> <br></br> <div>The problem builds on the ideas of fairness introduced in <a href="http://nrich.maths.org/public/viewer.php?obj_id=2150&amp;part=index">Domino Pick</a> and would benefit from discussion.</div> <br></br> <div>Of course this problem could also be tackled by the pupils themselves throwing the balls outside and data being collected for the whole class.</div> <br></br> <h3>Key questions</h3> <div>Why not find three balls so that you can try playing the game with a friend. What happens?</div> <div>Is it important that there is only red ball but two blue balls?</div> <div>How could you record what you have found out?</div> <div>Would it be helpful to use some red and blue counters for working out the different possibilities?</div> <br></br> <h3>Possible extension</h3> Those who found this easy could try this Stage $2$ problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=2150&amp;part=index">Tricky Track</a>.<br></br> <h3>Possible support</h3> Suggest trying with real balls first. Some red and blue counters might be helpful for working out the different possibilities.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">It might help to find three balls so that you can try playing the game with a friend. What happens?<br></br>Is it important that there is only red ball but two blue balls?<br></br></mdoxml> 2 5 1 0 0 0 0 Terry and Ali are playing a game with three balls. Is it fair that Terry wins when the middle ball is red? Decision Mathematics and Combinatorics Combinations Probability Theoretical probability Collecting Data Data collection Probability Experimental probability Information and Communications Technology Interactivities