A Chance to Win?
Problem
A Chance to Win printable worksheet
Imagine you were given the chance to win some money...
and imagine you had nothing to lose...
Imagine you arrive in a room where you are given £128 and six cards (3 red winning cards and 3 black losing cards).
You are asked to choose and lay the cards down, one at a time.
You can decide in which order to lay them down.
At each stage you must bet exactly half the money that you have available.
If you select and play a black card you lose the money you bet.
If you select and play a red card you receive double the money you bet
(ie. you get the money you bet back, plus that amount again, so if you bet £64 and win, your total will increase by £64).
If you end up with more money than you started with you get to keep the profit.
What's the best order for laying down the cards?
What will your strategy be when you are offered 4 or 5 red winning cards?
Draw some conclusions on what strategy to adopt and try to justify your findings.
We are indebted to Rob Eastaway for introducing us to this problem.
Getting Started
Imagine you start by choosing a black card and then choose two red cards:
| Your amount | Your bet | You chose | Result |
|---|---|---|---|
| £128 | £64 | black | - £64 |
| £64 | £32 | red | + £32 |
| £96 | £48 | red | + £48 |
Keep a record of the outcome from laying down the cards in different orders.
What do you notice?
How is your total amount of money affected by a win?
How is it affected by a loss?
Student Solutions
Tim explained clearly why you can never win:
It doesn't matter what order you play the cards!Each black card makes you lose half your money - which is the same as multiplying what you have by 0.5.
Each red card multiplies what you have by 1.5 since you win half of the cash you have.
As there are 3 reds and 3 blacks, the total amount you will finish with will be:
"starting amount" x 0.5 x 0.5 x 0.5 x 1.5 x 1.5 x 1.5 = starting amount x 0.42 (approx).
Since you can multiply in any order, you will always end up with the same amount! ( £54)
Charlie, Gemma, Griselda, Tom, Jason and Greg from Colyton Grammar School also worked out what was happening:
It does not matter in what order the cards are put down because the result of using a win card is that your money is multiplied by 1.5 and the result of you using a lose card is that your money is multiplied by 0.5.As multiplication is commutitive whatever order the cards are placed the results will be the same!
A lose followed by a win results in a return of 3/4 (0.5 x 1.5) of your stake.
This is with a ratio of 1:1 red:black.
Given the 6 cards the return is 27/64 of your stake.
In order to win, the ratio of red:black cards has got to be at least 1.71:1.
This is because in order to cancel the effect of one black card which is multiplying by 0.5 you need to multiply it by at least 2 as 0.5 x 2 = 1.
Since each red card multiplies by 1.5 you need to find the number (power) of 1.5's you need to multiply to get 2.
That is, what is x if $1.5^x$ is going to be greater than or equal to 2? (Answer: x has to be greater than or equal to 1.71)
Teachers' Resources
Why do this problem?
This problem improves students’ fluency in calculating proportion/percentage increases and decreases, and could be used to encourage students to use multiplicative methods rather than additive methods. It is also an opportunity for students to develop reasoning and mathematical arguments which explore and exploit the commutative property of multiplication.
There is the possibility of investigating novel equations with unknowns as powers.
Possible approach
This printable worksheet may be useful: A Chance to Win
We might begin by asking students to guess solutions, possibly prompted by questions such as ‘is it better to win early on?’ We could note any conjectures on the main whiteboard.
Next, students to try ordering the cards and calculate their winnings in pairs or small groups. Encourage them to record their findings logically. You could hand out manipulatives to represent the red and black cards.
Ask students to present their best sequence of cards. This is a chance for them to describe their thinking, including how it was guided by the initial conjectures. It will also ensure that everyone has understood the task and could showcase a variety of methods for calculating the final winnings.
The final winnings should be the same for every sequence presented, so you could challenge the students to find an order which gives a different total. If they can’t, why not? Once they’ve had a chance to develop their reasoning in small groups, you could have a class discussion about why the order of the cards will not affect the final winnings.
Next introduce the idea of additional winning cards. Some students might want some time to check that their conclusions still hold true. Others may be ready to immediately tackle the questions ‘how many winning cards do I need to make the game profitable?’, or ‘what happens it there is 1 losing card and $n$ winning cards?’. You could suggest that the students use a table to record their results.
You could bring the class together at the end to present and discuss their conclusions.
Key questions
Can you calculate your winnings for a particular sequence?
Do you notice anything about your results? Can you explain why this is?
How many winning cards do you need to make the game profitable?
How many winning cards do you need to ‘undo’ the effect of a losing card?
Possible support
To lead students towards seeing that the order of the cards does not matter, students might be provided with a table of suggested orders, and space to write their winnings. Calculators could be used if students are struggling numerically.
To help students express finding proportions/fractions/percentages as multiplication, you could use https://nrich.maths.org/2877 and/or https://nrich.maths.org/2877.
It is possible students will struggle with the concept of a ‘stake’; this might be illustrated through an example sequence of cards, or by first introducing a simpler game.
Possible extension
Win or Lose uses similar concepts. Students could also consider what happens if they are allowed to choose the amount they bet, or if they are given different odds on winning.