Probably ...

7245 /www/nrich/html/content/id/7245/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> This problem is designed for you to work on in a group of about four.<br></br> <br></br> You will need a set of <a href="/content/id/7245/ProbablyCards.doc">these six cards</a> for your group. Each card has a statement on it. These are the six statements:<br></br> <br></br> 6 is the hardest number to throw on a dice.<br></br> A game is fair if you play it properly.<br></br> I will see someone I know on the way home from school.<br></br> It always rains in the summer holidays.<br></br> If you buy lots of raffle tickets, you're sure to win a prize.<br></br> It's easier to get heads than tails when you flip a coin.<br></br> <br></br> The idea is for you to decide, as a group, whether you agree or disagree with each statement. Talk together about what you think and once you have made a decision, make a note of the reasons for your choice.<br></br> <br></br> If possible, discuss your group's reasons with the rest of the class.<br></br> <br></br> We would love to hear about your decisions, with your reasons of course!<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">We've received a large number of good responses to this problem.</span><br></br> <br></br> <span class="editorial">Alex, Chris, Alastair and Henry from St Peter's College and Emmy gave a comprehensive response to the problem:</span><br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: $6$ is the hardest number to throw on a dice</span>.<br></br> Response: $6$ is not the hardest number to throw on a dice, it has an equal chance of being thrown as all the other numbers from $1$-$6$<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: A game is fair if you play it properly</span>.<br></br> A game is not always fair if you play it properly. In sport, the umpires could favour a losing team because they were getting demolished. If the winning team is playing properly, then it is not always fair.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: I will see someone I know on the way home from school.</span><br></br> People do sometimes see people on the way home from school, it can happen. If you live right next to someone from your school, the chances are better than if you live $1000$ miles away. Also, if you catch a school bus, the chances are 100%.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: It always rains in the summer holidays.</span><br></br> It doesn't always rain in the summer holidays, but the chances are very high as it is a very long period.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: If you buy lots of raffle tickets, you're sure to win a prize</span>.<br></br> If you buy lots of raffle tickets you are not assured of winning a prize, it just increases your chances. The only way to assure yourself of winning a prize is cheating :) or buying all of the raffle tickets.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: It's easier to get heads than tails when you flip a coin.</span><br></br> Yes, it is easier to get heads rather then tails on a coin flip because the tails side of a coin is heavier than the heads side, this marginly increases you chances, but not by much. <br></br> <br></br> <span class="editorial">I am not sure about the response to the last statement ... What's your evidence for this, I wonder?!</span><br></br> <br></br> <span class="editorial">Gabriel and Oscar from St Saviours CE Primary School also sent a great response using interesting examples demonstrating their point:</span><br></br> <br></br> We disagree with all the statements.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: A game is fair if you play it properly</span><br></br> <br></br> In the England v Germany game in the World Cup 2010, Frank Lampard scored and everyone was playing properly, but the referee didn't see the goal so the goal wasn't allowed. This wasn't fair, but everyone was playing properly.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: 6 is the hardest number to throw on a dice.</span><br></br> There are an equal number of each number on the dice so there is a one in six chance of rolling any number.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: I will see someone I know on the way home from school.</span><br></br> If you were blind you wouldn't see anyone on your way home and if you were in a taxi and you fell asleep you wouldn't see anyone until you were at home.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: It always rains in the summer holidays.</span><br></br> It depends where you are living in the summer. For example if you lived in Libya, it wouldn't rain. Or if your holidays were only short, there would be a smaller chance of rain because there would be less days for it to rain on.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: If you buy lots of raffle tickets, you're sure to win a prize</span>.<br></br> There would be a bigger chance of winning but you couldn't be sure of anything.<br></br> <br></br> <span style="font-weight: bold; font-style: italic;">Statement: It's easier to get heads than tails when you flip a coin.</span><br></br> There is a 50% chance of getting either heads or tails because there is one of each. <br></br> <br></br> <h4><span class="editorial">Very well done to everyone - great reasoning.</span></h4> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Probably ...</h2> <br></br> This problem is designed for you to work on in a group of about four.<br></br> <br></br> You will need a set of <a href="/content/id/7245/ProbablyCards.doc">these six cards</a> for your group. Each card has a statement on it. These are the six statements:<br></br> <br></br> 6 is the hardest number to throw on a dice.<br></br> A game is fair if you play it properly.<br></br> I will see someone I know on the way home from school.<br></br> It always rains in the summer holidays.<br></br> If you buy lots of raffle tickets, you're sure to win a prize.<br></br> It's easier to get heads than tails when you flip a coin.<br></br> <br></br> The idea is for you to decide, as a group, whether you agree or disagree with each statement. Talk together about what you think and once you have made a decision, make a note of the reasons for your choice.<br></br> <br></br> If possible, discuss your group's reasons with the rest of the class.<br></br> <br></br> We would love to hear about your decisions, with your reasons of course!<br></br> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>Probability is an area which children can find difficult, largely due to the difference between experimental and theoretical probability. <a href="http://nrich.maths.org/7245&part=">This problem</a> is designed to get children talking freely about issues associated with probability. It could be used for you to assess children's understanding at the start or end of work on this topic. </div> <br></br> <h3>Possible approach</h3> <div>Divide the class up into groups of four to six and give each group one set of <a href="/content/id/7245/ProbablyCards.doc">these cards</a>. It might be that you want groups to choose a card at a time and discuss it altogether immediately. Alternatively, you could encourage each child to take one card from the set and to think about it on their own before discussing it with other members of the group.</div> <div> </div> <div>Give groups a suitable period of time to discuss each card. They must reach a consensus, deciding whether or not the group as a whole agrees with the statement. Each group could be given a large sheet of sugar paper on which to record their decision and reasons for each card.</div> <div> </div> <div>It will be important to allow plenty of time for the whole class to discuss their thoughts together. Listen out for those children who want to 'qualify' the statements further - this demonstrates higher-order thinking. Depending on their experience, you might expect groups to begin to quantify their reasons to the dice and coin statements. This final discussion has the potential to be very powerful as children often have the tendency to believe that mathematical probability has nothing to do with 'real-life'. This is your chance to deal with any misconceptions in a non-threatening way. </div> <br></br> <h3>Key questions</h3> <div>Tell me what you have discussed so far.</div> <div>How are you going to come to an agreement?</div> <div>Can you explain why you think that?</div> <div>How do you know? </div> <br></br> <h3>Possible extension</h3> <div>Some children might enjoy creating their own statements for others to discuss. You might like to encourage each group to make up one of their own and pass it to another group. </div> <br></br> <h3>Possible support</h3> <div>This activity presupposes that the children are used to working in groups. There may be disputes amongst group members but try to encourage them to sort these out themselves rather that you stepping in. Some children may need your support in constructing arguments and justifications. </div> <div> </div> <div> </div> <div><span style="font-style: italic;">You can read about some of the issues which might arise when teaching probability in</span> <a href="http://nrich.maths.org/7326&part=" style="font-style: italic;">this article</a><span style="font-style: italic;">.</span></div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">This problem is best tackled with a few other people so you can talk to each other about your ideas. <br></br> How will you convince someone else of your argument? <br></br></mdoxml> 5 3 0 1 0 0 0 Probably ... You'll need to work in a group for this problem. The idea is to decide, as a group, whether you agree or disagree with each statement. Probability Experimental probability Probability Theoretical probability Admin Upper primary mapping document