7244
/www/nrich/html/content/id/7244/
1
2011-02-01T00:00:01
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In a far-away land, the lottery consists of four balls numbered $1$
to $4$, which are placed in a bag.<br></br>
To enter, you choose one number.<br></br>
To win, your number must match the number that is drawn from the
bag.<br></br>
<br></br>
What is the chance of winning this lottery?<br></br>
<br></br>
The people running the lottery in this far-away land decide that it
is too easy to win. So, they change their lottery game.<br></br>
In the new lottery, there are still four balls numbered $1$ to $4$,
which are placed in a bag.<br></br>
Now, to enter, you choose two numbers.<br></br>
To win, your numbers must match (in any order) the two numbers that
are drawn from the bag.<br></br>
<br></br>
What is the chance of winning this new lottery?<br></br>
Have the organisers made it harder to win compared with their
original version?<br></br>
<br></br>
Can you create your own version of the lottery which would also be
harder to win than the first game? <br></br>
How do you know that your game is harder?<br></br>
<br></br>
You may like to use our lottery simulator to try out your
ideas:<br></br>
<br></br>
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<span class="editorial">We have received many good responses for
this question. It is interesting to see all the different lottery
rules you have come up with.</span><br></br>
<br></br>
<span class="editorial">Rosalind from Ricards Lodge High School
made a good start on the first question</span>:<br></br>
<br></br>
There are four balls and you must choose one, so it's 1 ball which
you choose out of the total of 4 balls that can be chosen. So
the chance is 1:4.<br></br>
<br></br>
<span class="editorial">Kennard Brentfield Primary
School also correctly answered this part of the
question.</span><br></br>
<br></br>
<span class="editorial">Another student from Ricards Lodge gave a
nice complete solution to the questions:</span><br></br>
<br></br>
My method of finding out what the probability is of winning the new
lottery, the old lottery game and if it is actually harder to win
than the old one is this:<br></br>
<br></br>
The old lottery game:<br></br>
In the old lottery game the probability of winning is
$\frac{1}{4}$. This is because there are four balls and you
pick out one. Therefore you have a chance of $1$ in $4$.<br></br>
<br></br>
The new lottery game:<br></br>
The probability of winning the new lottery game is $\frac{1}{6}$.
This is because there are six options you could have:<br></br>
$1$ and $2$, $1$ and $3$, $1$ and $4$, $2$ and $3$, $2$ and
$4$, $3$ and $4$.<br></br>
You can only pick one pair out of the six options so the
probability is $1$ in $6$. Therefore it is harder to win the new
lottery game than the old one.<br></br>
<br></br>
If I were going to make a newer even harder lottery game, this
would be my method:<br></br>
There would be four balls and you can pick out any two. Then
if your balls were picked out in the same order as you picked out
yours then you win!<br></br>
This is harder because the balls have to be picked out in the
correct order making it a lot more difficult to win! <br></br>
<br></br>
<span class="editorial">Daisy, also from Ricards Lodge submitted an
excellent solution with similar methods. She also investigated the
chances of winning if three balls are chosen from four in the
bag:</span><br></br>
<br></br>
The chance of you winning the lottery in Far away land is
$\frac{1}{4}$ because there are only four balls to pick from.<br></br>
When they made it slightly harder the chance of winning was
$\frac{1}{6}$. The possible options you could choose were (without
repeating yourself):<br></br>
$1$ and $2$, $1$ and $3$, $1$ and $4$, $2$ and $3$, $2$ and $4$,
$3$ and $4$.<br></br>
There were only six possible outcomes, therefore your chance of
winning was $\frac{1}{6}$.<br></br>
<br></br>
Using four balls the probability of you winning does not get any
harder then $\frac{1}{6}$ unless you need a pattern.<br></br>
For example if you needed to match with three balls then there
are fewer outcomes, in fact only four which brings you back to
the probability of $\frac{1}{4}$.<br></br>
<br></br>
<span class="editorial">Alistair from Charters School, Krystof
from Uhelny Trh, Prague also sent us good solutions to
the first parts of the question.</span><br></br>
<br></br>
<span class="editorial">Katy from Ricards Lodge told us her idea to
make the lottery harder, by creating more possible
outcomes:</span><br></br>
<br></br>
My idea is to have $150$ national lottery balls, only one of them
picked. There are $150$ possible outcomes, which is greater than
$6$ or $4$ possible outcomes.<br></br>
<br></br>
<span class="editorial">Ali from Riversdale Primary also shared how
he arrived at his method to make winning more
difficult:</span><br></br>
<br></br>
I tried to make it harder by picking three balls, but the
probability of winning was $1$ in $4$ again, because there is only
one ball left in the bag.<br></br>
<br></br>
To make it harder you have to increase the number of balls. I added
one more and found that the chance of winning is $1$ in $5$ if you
have to pick one or four balls, or $1$ in $10$ if you have to pick
two or three balls.<br></br>
<br></br>
<span class="editorial">Alice, Oliver, Michael, Elsa and Rosie from
the Extension Maths group at St Nicolas C of E Junior School,
Newbury also answered the first two parts of the problem
correctly. Alice and Oliver created their own, harder version
of the lottery:</span><br></br>
<br></br>
Our lottery uses five balls, numbered $1$ to $5$, placed in a
bag and you must choose two numbers. Your numbers must match, in
the same order, the balls drawn from the bag.<br></br>
We listed all the possibilities:<br></br>
$1$ and $2$, $1$ and $3$, $1$ and $4$, $1$ and $5$, $2$ and $1$,
$2$ and $3$, $2$ and $4$, $2$ and $5$, $3$ and $1$, $3$ and $2$,
$3$ and $4$, $3$ and $5$, $4$ and $1$, $4$ and $2$, $4$ and $3$,
$4$ and $5$, $5$ and $1$, $5$ and $2$, $5$ and $3$, $5$ and
$4$.<br></br>
That makes $20$ and they are all equally likely. We know our
lottery is harder to win because there are more possibilities. The
chances of winning are $\frac{1}{20}$ not $\frac{1}{4}$.<br></br>
<br></br>
Elsa and Rosie chose a simpler version: <br></br>
Number of balls: nine Pick: one <br></br>
Answers: $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ <br></br>
Draw: one ball <br></br>
Chance of winning: $\frac{1}{9}$ <br></br>
There is one ball picked out of the bag and there are
nine balls that have an equally likely chance of being picked.
So $9$ will be the denominator. There is one draw so $1$ will be
the numerator, so that will equal $\frac{1}{9}$ which means it is
harder.<br></br>
<br></br>
<span class="editorial">Benjamin from Flora Stevenson Primary
School also worked out the problem using the same
thinking.</span><br></br>
<br></br>
<span class="editorial">Well done to all of you who sent in your
solutions.</span><br></br></mdoxml>
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<h2>Winning the Lottery</h2>
<br></br>
In a far-away land, the lottery consists of four balls numbered $1$ to $4$, which are placed in a bag.<br></br>
To enter, you choose one number.<br></br>
To win, your number must match the number that is drawn from the bag.<br></br>
<br></br>
What is the chance of winning this lottery?<br></br>
<br></br>
The people running the lottery in this far-away land decide that it is too easy to win. So, they change their lottery game.<br></br>
In the new lottery, there are still four balls numbered $1$ to $4$, which are placed in a bag.<br></br>
Now, to enter, you choose two numbers.<br></br>
To win, your numbers must match (in any order) the two numbers that are drawn from the bag.<br></br>
<br></br>
What is the chance of winning this new lottery?<br></br>
Have the organisers made it harder to win compared with their original version?<br></br>
<br></br>
Can you create your own version of the lottery which would also be harder to win than the first game? <br></br>
How do you know that your game is harder?<br></br>
<br></br>
You may like to use our lottery simulator to try out your ideas:<br></br>
<br></br>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/7244&part=">This problem</a> offers an engaging context in which to develop children's understanding of experimental and theoretical probability. The interactivity will give pupils a 'feel' of the situation through the experimental data, then they can be encouraged to calculate theoretical probabilities by working systematicallly and listing all
possible winning combinations.</div>
<br></br>
<h3>Possible approach</h3>
<div>Ideally, it would be good to have access to the Lottery Simulator interactivity as you introduce and work on this problem. If this is not possible, you could simulate the lottery yourself by having numbered balls or digit cards in a bag. You could pick out the 'winning' number yourself and record the results somewhere for all to see. </div>
<div> </div>
<div>To begin with, you could set up the interactivity so that only four balls are available in the 'Number Tumbler' and 'your ticket' has just one number. Explain the way this lottery works to the group and invite suggestions for the number to choose on your ticket. You could click on the subtraction sign in the 'Lottery Simulator' panel to select just one draw and then
click the 'Simulate lottery draws' button to see what happens. You could simulate a draw a few times.</div>
<div> </div>
<div>Once the group has got a feel for what the interactivity is doing, you could pose a few questions, such as:</div>
<div>What is the chance of winning our lottery?</div>
<div>How many times would you expect to win in ten draws?</div>
<div>How many times would you expect to win in twenty draws?</div>
<div>How many times would you expect to win in fifty draws? </div>
<div>How many times would you expect to win in one hundred draws?</div>
<div>Give children time to work on these questions in pairs, making it clear that you will be expecting them to explain their thinking.</div>
<div> </div>
<div>After a suitable period, bring the whole class together to share their ideas and to ask them to justify their predictions mathematically. For example, some children might say that there are four possible winning numbers ($1$, $2$, $3$ or $4$) and if you have chosen one of them on your ticket, the chance of you winning is $1$ out of $4$ or a quarter or
$25$%. </div>
<div> </div>
<div>You could use the interactivity to simulate ten/twenty/fifty etc draws. Did the simulations match their predictions? This is a good chance to bring up the idea that the higher the number of draws, the more likely the experimental data (i.e. the simulator) will match the theoretical data (i.e. one in four).</div>
<div> </div>
<div>You can encourage the group to investigate the modified version of the lottery in a similar way. This time, it is more of a challenge to list all the possible draws.</div>
<br></br>
<h3>Key questions</h3>
<div>What number/s could be drawn?</div>
<div>What are all the possible draws? </div>
<div>How do you know you have got them all?</div>
<div>How can you tell which version of the lottery is easier/harder to win?</div>
<br></br>
<h3>Possible extension</h3>
<div>Encouraging learners to find their own ways to make it more difficult to win the lottery is a never-ending task! You may be surprised to observe those who stick to situations which they feel comfortable analysing and those who push themselves a bit further. </div>
<br></br>
<h3>Possible support</h3>
<div>It might be helpful for learners to have access to the interactivity in pairs.</div>
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What number/s could be drawn?<br></br>
What are all the possilble draws?<br></br>
How do you know you have got them all?<br></br>
How can you tell which version of the lottery is easier/harder to
win?<br></br>
You could use the interactivity to have a go at playing the lottery
many times.<br></br>
<br></br>
The following might help you use the interactivity:<br></br>
<br></br>
Use the +/- buttons in the top left hand corner to increase or
decrease the number of balls in the Number Tumbler. For example, to
simulate the UK National Lottery, increase the number of balls to
49. <br></br>
Use the +/- buttons by 'Your Ticket' to increase or decrease the
number of spaces on your lottery ticket. For example, to simulate
the UK National Lottery, increase the number of spaces to 6. <br></br>
Then click on the numbers in the Number Tumbler to select which
numbers you would like on your lottery ticket. <br></br>
At the bottom of the screen, use the +/- buttons to choose how many
draws will be simulated. <br></br>
Once the simulator is set up, click the 'Simulate lottery draws'
button to run the lottery your desired number of times.<br></br>
The 'Matches black' button marks every time one of the numbers
from your ticket was drawn. The 'Sort matches' button rearranges
the draws with all your winning tickets together. <br></br></mdoxml>
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<mdoxml version="1.0">Ways to win new version:<br></br>
1, 2<br></br>
1, 3<br></br>
1, 4<br></br>
2, 3<br></br>
2, 4<br></br>
3, 4<br></br></mdoxml>
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Winning the Lottery
Try out the lottery that is played in a far-away land. What is the
chance of winning?
Probability
Experimental probability
Probability
Theoretical probability
Using, Applying and Reasoning about Mathematics
Working systematically
Information and Communications Technology
Interactivities
Admin
Upper primary mapping document