## Coin toss game

By David Shaw, 15 October 2013

Here’s a fun game that will help you explore probability. With a bit of thought you’re more likely to win. Are you smart enough to work it out?

### You will need

Rule lines every five centimetres on a sheet of cardboard.

- A large sheet of white cardboard
- Long ruler
- Permanent marker
- 20c, $1 and $2 coins
- Old table (may get scratched)
- Masking tape
- Paper and pen for recording results

### What to do

- Measure along one side of the cardboard and make a mark every 5 cm. Then mark every 5 cm along the opposite side, making sure you start measuring from the same end of the cardboard.
You should end up with a grid of squares.

- Mark every 5 cm on the two remaining sides, once again remembering to measure from the same end of the cardboard.
- Join each dot to the corresponding dot on the other side of the cardboard. You should end up with a square grid pattern on your cardboard.
- Tape the cardboard down on a table.
- Stick a line of masking tape on the ground 1.5 m from the table.
- Stand behind the sticky tape and throw coins on to the cardboard sheet. If the coin lands on the sheet without touching any of the lines, you win! If it touches a line, you lose. If it falls off the cardboard, you can throw again.
If your coin touches a line, you lose. if it doesn’t cross a line, you win!

- Play this game 20 times with a 20c coin, 20 times with a $1 coin and 20 times with a $2 coin. Write down the number of times you won with each type of coin. Which coin is easiest to win with?

### What’s happening?

In this game, a smaller coin is more likely to win. Since you are throwing from far away, you don’t have much control over where the coin lands. Whether it ends up on a line should be more down to luck than skill. If it is entirely luck, you can do a simple bit of geometry to see the chances of winning or losing.

Look at one of your coins. It’s pretty close to being a circle, so if the centre is close to a line, then the coin will touch it. To work out the chances of winning, draw strips as wide as the coin centred on each of the lines on the cardboard. If the centre of the coin is in a strip, the coin must overlap a line and you have lost. If the centre of the coin is not in a strip, then the coin does not overlap a line, and you have won.

You can colour the cardboard to show how likely you are to win.

If you colour the strips, you can easily see the areas where you win and the areas where you lose. By measuring the areas inside and outside the strips, you can work out the probability that you will win. You could make a different sheet for each size of coin to work out how much easier it is with a $2 coin!

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