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'Winning the Lottery' printed from https://nrich.maths.org/
We have received many good responses for
this question. It is interesting to see all the different lottery
rules you have come up with.
Rosalind from Ricards Lodge High School
made a good start on the first question:
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.
Kennard Brentfield Primary
School also correctly answered this part of the
question.
Another student from Ricards Lodge gave a
nice complete solution to the questions:
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:
The old lottery game:
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 $4$.
The new lottery game:
The probability of winning the new lottery game is $\frac{1}{6}$.
This is because there are six options you could have:
$1$ and $2$, $1$ and $3$, $1$ and $4$, $2$ and $3$, $2$ and
$4$, $3$ and $4$.
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.
If I were going to make a newer even harder lottery game, this
would be my method:
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!
This is harder because the balls have to be picked out in the
correct order making it a lot more difficult to win!
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:
The chance of you winning the lottery in Far away land is
$\frac{1}{4}$ because there are only four balls to pick from.
When they made it slightly harder the chance of winning was
$\frac{1}{6}$. The possible options you could choose were (without
repeating yourself):
$1$ and $2$, $1$ and $3$, $1$ and $4$, $2$ and $3$, $2$ and $4$,
$3$ and $4$.
There were only six possible outcomes, therefore your chance of
winning was $\frac{1}{6}$.
Using four balls the probability of you winning does not get any
harder then $\frac{1}{6}$ unless you need a pattern.
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 $\frac{1}{4}$.
Alistair from Charters School, Krystof
from Uhelny Trh, Prague also sent us good solutions to
the first parts of the question.
Katy from Ricards Lodge told us her idea to
make the lottery harder, by creating more possible
outcomes:
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.
Ali from Riversdale Primary also shared how
he arrived at his method to make winning more
difficult:
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.
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.
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:
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.
We listed all the possibilities:
$1$ 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
$4$.
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 $\frac{1}{4}$.
Elsa and Rosie chose a simpler version:
Number of balls: nine Pick: one
Answers: $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$
Draw: one ball
Chance of winning: $\frac{1}{9}$
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.
Benjamin from Flora Stevenson Primary
School also worked out the problem using the same
thinking.
Well done to all of you who sent in your
solutions.