# Heads and tails

This activity is about tossing coins. A coin always has two faces, usually called:

Heads

and Tails

When you toss a coin and let it land on a flat surface (like the floor or a table top), it could land in three ways:

- Heads up, which we abbreviate to Heads, or just H.
- Tails up, which we abbreviate to Tails, or just T.
- On its edge. Well, you might think that's impossible, but it's not. I've actually seen it happen! However, it is very unlikely to happen, so we usually just ignore it.

So there are really just two possibilities: H and T.

## How to toss a coin?

Tossing a coin can be quite tricky if you don't know how. You need to place the coin on your thumb and index finger and flick it into the air using your thumb:

If you find that too difficult, you could put the coin in a small cup and shake it around before releasing the coin.

You may have seen sports events where one of the captains tosses a coin, catches it with one hand and then places it on the back of their other hand. We won't do that. Just let the coin fall onto a flat surface (like the floor or a table top).

## Experiment 1: Toss a single coin 100 times

For this experiment you need just one coin and need to toss it one hundred times. Record your results in a table, like this:

Outcome | Tally | Frequency |

Heads | ||

Tails | ||

Total = 100 |

If you've forgotten how to do a tally, go to the page Tally Marks

Now do a Bar Graph to illustrate you results. Did you find that the two bars were almost the same height? You should have done, as long as your coin was a fair one.

## What is a fair coin?

A fair coin is one that has no bias which means Heads is not more likely to occur than Tails, and vice versa.

It's unlikely that you will have a coin that is biased. That would mean the coin will have been weighted so that it's more likely to land on one face than the other.

So when you toss a fair coin 100 times, you should expect to get roughly 50 Heads and 50 Tails. That is because Heads and Tails are equally likely. The probabilities of each event - Heads and Tails - are both equal.

Because they are equal, they are both given a probability of ½.

So:

- Probability of Heads = ½
- Probability of Tails = ½

We abbreviate this to: P(H) = ½ and P(T) = ½

But you're probably going to tell me that you didn't get exactly 50 of each. Does that make your coin biased?

Not at all.

We have to be careful to distinguish between the theoretical values (50 Heads and 50 Tails) and the experimental results (which may not be exactly 50 of each, but close).

Note: If you were to toss your coin 1,000 times, you would probably get a much better result. You can try it if you like.

## Experiment 2: Toss two coins together 100 times

For this experiment you need two coins. The two coins could be of the same denomination, but it might be a good idea to use two different ones (e.g. one $1 coin and one 50c coin). Why? Well, answer this question first: How do you toss two coins together? You don't have to toss them both at the same time; one after the other will do. Or you can use the cup method and release both coins at the same time. It's not the way you toss the coins, but it's the result of the two coins that really matters.

### How many possible results are there?

- Both coins could land Heads up - that's one result, HH
- Or both coins could land Tails up - a second result, TT
- Or one could land Heads and the other Tails - a third result, HT

Is that all? The answer is: Yes and No

Think about this question: Is the outcome 'A Head on the $1 coin and a Tail on the 50c coin' the same as the outcome 'A Tail on the $1 coin and a Head on the 50c coin'?

Of course they are different. One is HT, the other is TH. But they can be bracketed together as 'One Head and one Tail'.

That's why it's a good idea to use two coins of different denominations, so that you can see the difference between 'A Head on the $1 coin and a Tail on the 50c coin' and 'A Tail on the $1 coin and a Head on the 50c coin'.

So we could record these results:

Outcome | Tally | Frequency |

Two Heads | ||

One Head and one Tail | ||

Two Tails | ||

Total = 100 |

But we can gather **more information** by recording this way:

Outcome | Tally | Frequency |

HH | ||

HT | ||

TH | ||

TT | ||

Total = 100 |

Go ahead and record the results that way!

Now draw a Bar Graph.

What sort of results did you get this time? Can you work out what the theoretical probabilities are? Write them down:

- P(HH) =
- P(HT) =
- P(TH) =
- P(TT) =

- P(Two Heads) =
- P(One Head and one Tail) =
- P(Two Tails) =

How close were your experimental results to what you should expect in theory? How could you get a better result?

## Experiment 3: Toss one coin 100 times and record the runs

What is a run?

A run of heads could be:

- HH - a run of length two
- HHH - a run of length three
- HHHH - a run of length four
- etc

Similarly TTTTTT is a run of tails of length 6.

You can also have a run of length one - either H or T - but as soon as you toss the opposite type, your run ends.

For the purposes of this experiment we are not going to distinguish between Heads and Tails, except to recognize when you change from one to the other and the run ends. So, for example HHH and TTT will both be counted as runs of length 3.

Now do the experiment and record your results in a table as follows:

Outcome | Run length | Tally | Frequency |

H or T | 1 | ||

HH or TT | 2 | ||

HHH or TTT | 3 | ||

HHHH or TTTT | 4 | ||

Total = 100 |

Until you do the experiment, you won't know what the longest run length will be, so you might need to add some more rows to your table.

Illustrate your results with a Bar Graph.

What shape did you get for your bar graph?

Are all outcomes equally likely this time?

Can you explain why?