1. ## Coin Board Game

The squares show a 10x10 design for a ‘roll a coin’ board at a local fair. Thedesign is repeated throughout the board. The squares are each of length 5 cm andthe only coins used are 2 cents of diameter 2.5 cm. Show that the probability that a coin of this diameter will soon come to rest nottouching the side of a square is 0.25.Coins are rolled on to the board and a prize is won if the coin comes to rest nottouching a line. The prize is either the amount, in cents, written on the squareor, if the square is blank, the coin is returned. Coins which come to rest on a lineare losers.Would this game make a profit or a loss and what fraction of the takings could beexpected to be given out as prizes?Make a 10x10 design for a ‘roll a coin’ board so that all the takings can beexpected to be given back as prizes, using at least one of each of 50c, 20c, 10c and 5c squares.

I don't understand how to show the probability as 0.25 and how the length of the squares affect it. I'm guessing that it is less likely that you would get a prize since there are more squares that are blank... Are there more than one possibility to make the design?

2. ## Re: Coin Board Game

Are you a coin board operator in some caboose at country fairs,
or is this a problem given out by your math teacher?

3. ## Re: Coin Board Game

Originally Posted by DenisB
Are you a coin board operator in some caboose at country fairs,
or is this a problem given out by your math teacher?
They were given by my teacher about probability. Can anyone help to solve this problem? I got the first part of the problem, where you can use the area of the coin compared to the area of the square and get the percentage (if wrong please correct me) , but I don't get the second question. How do you know whether it would be a gain or not if you don't know the number of throws? For the third problem, wouldn't you just fill the board with the numbers 5,10,20 and 50 to get prizes in all throws?

4. ## Re: Coin Board Game

Take a look at this picture.

The striped area is the loci of points that the center of the coin can take and not hit the lines of the square.

If the square has sides length $s$ and the coin is of radius $r$

then the area of this loci is $(s - 2r)^2$

Assuming a uniform probability of where coin centers will fall,

the probability of the coin not hitting a line is thus

$P[\text{coin doesn't hit a line}] = \dfrac{(s-2r)^2}{s^2}$

substituting in your numbers we have

$P[\text{coin doesn't hit a line}] = \dfrac{(5-2(1.25))^2}{5^2} = 0.25$

do you have questions on the rest of it?

5. ## Re: Coin Board Game

if you chug through all this you end up with the following distribution of gains

$P[G] = \begin{cases} -2 &0.75 \\ 0 &(0.25)(0.9) \\ 3 &(0.25)(0.04) \\ 8 &(0.25)(0.03) \\ 18 &(0.25)(0.02) \\ 48 &(0.25)(0.01) \end{cases} = \begin{cases} -2 &0.75 \\ 0 &0.225 \\ 3 &0.01 \\ 8 &0.0075 \\ 18 &0.005 \\ 48 &0.0025 \end{cases}$

note the gains are the prize less the 2c you spent to play. If you successfully land, and in a blank space, you get the coin back, otherwise you just get the prize.

you can find the expectation of this distribution easily enough

$E[G] = -1.2$

So this board would make a profit of $1.2c$ per $2c$ roll on average

6. ## Re: Coin Board Game

For the third part of the question, would I have to put more numbers in the board until the there is a profit higher than 2c?

7. ## Re: Coin Board Game

Originally Posted by YLEE
For the third part of the question, would I have to put more numbers in the board until the there is a profit higher than 2c?
No. Design it so profit is zero. I.e.

$E[G]=0$

8. ## Re: Coin Board Game

But if all the takings need to be returned as prizes, wouldn't that mean you would have to make more profit?

9. ## Re: Coin Board Game

Originally Posted by YLEE
But if all the takings need to be returned as prizes, wouldn't that mean you would have to make more profit?
if all the takings are returned as prizes you make zero profit.

10. ## Re: Coin Board Game

Ok, my calculations say that in the design, you can add 4 20s and 6 50s in the board for the profit to be 0. Are there other possibilities than this one? Also, I was wondering whether the location where I put them could affect the chance that the person would get the coin in the squares: for example, if two numbers were right next to each other? I was thinking about this because in the original table the squares with numbers never touch.

11. ## Re: Coin Board Game

Originally Posted by YLEE
Ok, my calculations say that in the design, you can add 4 20s and 6 50s in the board for the profit to be 0. Are there other possibilities than this one? Also, I was wondering whether the location where I put them could affect the chance that the person would get the coin in the squares: for example, if two numbers were right next to each other? I was thinking about this because in the original table the squares with numbers never touch.
they want at least 1 each of 5, 10, 20, 50.

We have no information on how a coin might react when it rolls so we can only assume a completely uniform distribution on where the center of the coin might end up.

So how you distribute the prize squares will have no effect.

12. ## Re: Coin Board Game

For the part where they say "what fraction of the takings could be expected to be given out as prizes?", should I put the probability that a coin will land on a square with a numbers as a fraction or the expected value as a fraction?