Place Your Marker on the Circle Marked (Mouse Starts Here)

Place Your Marker on the Circle Marked (Mouse Starts Here)

Cat and Mouse

Cat and mouse is a game played on the game board above with a dice and a marker.

How to play:

-Place your marker on the circle marked (mouse starts here)

-Roll the dice and follow either the odd (O) path or the even (E) path depending on the number rolled.

-Continue until the mouse either eats the cheese or gets eaten by the cat.

Investigation – Section 1

Question 1)

Play the game 10 times and record the outcome of each game.

Question 2)

Does the game seem to be fair? If not who does it favour the cat or the mouse?

Question 3)

If you played the game again would you expect to get the same results? Why/Why not?

Question 4)

How could we test further to see if the game is fair?

Question 5)

Collect results from other people doing the same project and continue to play the game until you have conducted 100 trials.

Question 6)

Over all how many times did the cat win? How many times did the mouse win?

Question 7)

Write your answers to question 6 as fractions, decimals and percentages.

Question 8)

How do your results to question 6 compare with your initial 10 games?

Question 9)

If we played the game 1000 times how often would you expect the cat to win? How often would the mouse win?

Question 10)

If we played the game 1,000,000 times would the cat or the mouse win more often?

Investigation - Section 2

In section 1, we investigated the fairness of the game board by playing the game a large number of times to get more accurate results. However there are a number of limitations to this method.

Question 1)

Name two limitations in determining the fairness of the game board using repeated trials? Can you think of any better ways to determine the fairness of the game?

One different approach to testing the fairness of the game board is to play many games at the same time and use our knowledge of probability instead of rolling the dice.

Question 2)

If you started with 32 mice all entering at the same time, how many would you expect to go down each path? (Think about the odds of a single mouse going down either path)

Question 3)

How many mice would end up at the circle labelled A? How many would end up at circle B?

Question 4)

We are going to just look at circle A for a moment. If all the mice currently at circle A left how many would go to the cat and how many would go to circle C?

Question 5)

If we all the mice at circle B left at the same time how many would go to circle C, and how many would go to circle D?

Question 6)

If all the mice at circle C now left how many would go to the cheese and how many would go to circle D?

Question 7)

Finally if all the mice at circle D left how many would go to the cheese and how many would go to the cat?

Question 8)

How many mice in total went to the cat and how many went to the cheese? Is the game fair or unfair?

We have just shown that the game is indeed fair and we never even picked up the dice! This process is often called “finding the general solution to a problem”, and involves removing the chance element from the situation and only working with the probabilities. It is often an extremely useful way to analyse situations of chance. Remember if we had played the game 32 times using the dice, would could have had the mouse going to the cheese every time, or the cat every time or any combination in between. This method allows us to determine what we would expect to happen.

The example above was guided because we started off with 32 mice, a number which could be repeated halved easily. But what would we have done if we had tried to start with 33 mice? How would you divided a mouse in half? This means that this method can at times be flawed, however, we can change it so it works all the time, we just have to use decimals.

Let’s look at the example below

All the mice start off at the beginning so the probability of the mice being there is 1

Step 1 – Half the mice go to A and half the mice go to B (shown in the blue), so the probability of a mouse being at A or B is 0.5

Step 2 – Half the mice at A go to the cat (0.25) and the other halve go to C (0.25)

Step 3 – Half the mice at B (0.25) also go to C, whilst the other half got to D

Step 4 – That now means that we have half the mice at C (0.25 +0.25 = 0.5). Half of these go to the cheese (0.25) and half go to D (0.25)

Step 5 – That now means that we have half the mice at D (0.25+0.25=0.5). Half of these go to the cat (0.25) and half to the cheese (0.25)

Step 6 – we can now see if we add up the mice at the cat and the cheese then half are at each (0.5) so the probability of getting to the cheese or the cat is 0.5

The advantage of the above method is that we don’t’ have to try and find a number that we can always divide they was we need to and at the end we can simply read what the probability of getting to the cat or cheese is.

Section 3 – design your own game board

Now it is time to create your own game board using some of the principals above. Below are the guidelines for creating your game board.

-There must be at least 4 other circles on top of the starting circle and the cat and cheese circle.

-You must come up with another way to split your mice, not just odds and evens (or any other way for it to be 50:50)all the time. For example it could be if you roll a 1 or 2 go down one path 3,4,5,6 go down another.

-The way you split your mice doesn’t always have to be the same, so you could go for odds and evens at one circle, 1 or 2 and 3,4,5 or 6 for another, and another circle with 3 paths coming off which are 1 or 2, 3 or 4, 5 or 6.

-You need to make your paths directional like on the original game board.

-Your game MUST be fair. (this is the hardest part)

You must show using one of the methods from section 2 that your game is indeed fair.