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.
Age
7 to 11
Challenge level
Problem
This problem is designed for you to work on in a group of about four.
You will need a set of these cards for your group. Each card has a statement on it. These are the six statements:
$6$ is the hardest number to throw on a dice.
A game is fair if you play it properly.
I will see someone I know on the way home from school.
It always rains in the summer holidays.
If you buy lots of raffle tickets, you're sure to win a prize.
It's easier to get heads than tails when you flip a coin.
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.
If possible, discuss your group's reasons with the rest of the class.
We would love to hear about your decisions, with your reasons of course!
You will need a set of these cards for your group. Each card has a statement on it. These are the six statements:
$6$ is the hardest number to throw on a dice.
A game is fair if you play it properly.
I will see someone I know on the way home from school.
It always rains in the summer holidays.
If you buy lots of raffle tickets, you're sure to win a prize.
It's easier to get heads than tails when you flip a coin.
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.
If possible, discuss your group's reasons with the rest of the class.
We would love to hear about your decisions, with your reasons of course!
Getting Started
This problem is best tackled with a few other people so you can
talk to each other about your ideas.
How will you convince someone else of your argument?
How will you convince someone else of your argument?
Student Solutions
We've received a large number of good responses to this problem.
Alex, Chris, Alastair and Henry from St Peter's College and Emmy gave a comprehensive response to the problem:
Statement: $6$ is the hardest number to throw on a dice.
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 $1$-$6$
Statement: A game is fair if you play it properly.
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.
Statement: I will see someone I know on the way home from school.
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%.
Statement: It always rains in the summer holidays.
It doesn't always rain in the summer holidays, but the chances are very high as it is a very long period.
Statement: If you buy lots of raffle tickets, you're sure to win a prize.
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.
Statement: It's easier to get heads than tails when you flip a coin.
Yes, it is easier to get heads rather than tails on a coin flip because the tails side of a coin is heavier than the heads side, this marginally increases your chances, but not by much.
I am not sure about the response to the last statement ... What's your evidence for this, I wonder?!
Gabriel and Oscar from St Saviours CE Primary School also sent a great response using interesting examples demonstrating their point:
We disagree with all the statements.
Statement: A game is fair if you play it properly
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.
Statement: 6 is the hardest number to throw on a dice.
There is an equal number of each number on the dice so there is a one in six chance of rolling any number.
Statement: I will see someone I know on the way home from school.
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.
Statement: It always rains in the summer holidays.
It depends on 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 fewer days for it to rain on.
Statement: If you buy lots of raffle tickets, you're sure to win a prize.
There would be a bigger chance of winning but you couldn't be sure of anything.
Statement: It's easier to get heads than tails when you flip a coin.
There is a 50% chance of getting either heads or tails because there is one of each.
Charlotte, Charlie, Harvey & Liam (Group 1) and Liv, Archie, Dominic & Tom (Group 2) Raquel, Miles & Tyler (Group 3) from Kineton Primary School sent in the following:-
Group 1
6 is the hardest number to throw on a dice.
It has the same amount of % chance to get a 6 as a 1 and all the other numbers because the chances are all equal. It's just about luck!A game is fair if you play it properly.
Yes, because if you cheat it means you are not playing the game fairly, although some people are faster than others, eg in tag, which can make the game seem unfair.I will see someone I know on the way home from school.
Yes because if you live in the same place as your school you are more likely to see someone you know on the way home.It always rains in the summer holidays.
Sometimes because the holidays are spread over 6 weeks; there is a 1 in 42 chance there will be a rainy day. In the UK it frequently rains it's rare to have 42 days of sunshine.If you buy lots of raffle tickets, you're sure to win a prize.
Yes because the more tickets you have bought the more chances you have of winning a prize. It is not guaranteed but there is a bigger chance.It's easier to get heads than tails when you flip a coin.
No, because there is a 50% chance between them both. It comes down to luck!Group 2
6 is the hardest number to throw on a dice.
No because there is a 1 in 6 chance of rolling any number. It's totally random which number you will roll.A game is fair if you play it properly.
No because it depends on what type of game it is...eg cheater's monopoly will have different rules! It's unfair to cheat!I will see someone I know on the way home from school.
Yes because you usually see them on the playground when you leave, it depends on where you live compared to other people as to who you might see.It always rains in the summer holidays.
No because you could live in Egypt or the Equator where it doesn't rain a lot and have no rain. It probably will rain because we are not close to the Equator and 6 weeks is a long time for it not to rain in the UK.If you buy lots of raffle tickets, you're sure to win a prize.
Yes but it depends on how many raffle tickets you buy. If you buy 1 there is a tiny chance and if you buy all of them you are guaranteed to win a prize. It depends also on how many prizes there are and how many tickets there are altogether.It's easier to get heads than tails when you flip a coin.
It's a 50:50 chance as you can't make the coin land on heads or tails if you want it to.Group 3
6 is the hardest number to throw on a dice.
The 6 on a dice is quite rare because it's all on your luck and it's a 16.6% chance of getting a 6 (1 in 6 chance)A game is fair if you play it properly.
Yes because you are not playing it properly it counts as cheating an that is breaking the rules and means it is not fair.I will see someone I know on the way home from school.
It depends on how close you live to someone you know and whether you are walking or going in the car. You are likely to see someone you know while walking as you have more time to bump into someone you know living in the same place as your school.It always rains in the summer holidays.
It depends on which country you live in and the recent weather you have had in that 6 week holiday.If you buy lots of raffle tickets, you're sure to win a prize.
Yes because the more tickets you have the bigger the chance of winning a prize. It also depends on how many you buy out of all of the tickets. 3 out of 1,000 is a small chance, yet 500:1000 is a bigger chance to win.It's easier to get heads than tails when you flip a coin.
It depends on your luck! It's a 1 in 2 chance of getting heads or tails.Some very good ideas, very well done to everyone - great reasoning.
Teachers' Resources
Why do this problem?
Probability is an area which children can find difficult, largely due to the difference between experimental and theoretical probability. This problem 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.
Possible approach
Divide the class up into groups of four to six and give each group one set of these cards. 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.
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.
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.
Key questions
Tell me what you have discussed so far.
How are you going to come to an agreement?
Can you explain why you think that?
How do you know?
Possible extension
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.
Possible support
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.