Experiments and the long run

What if we cannot count the outcomes?

For a die, we can count. For a drawing pin that falls with its point up or down, the two outcomes are not equally likely, and no counting tells us the chance. We can only try it.

Drop the pin $200$ times. If it lands point up $130$ times, then the relative frequency of point up is

This is our estimate of the probability. It is not exact, but it is the best we have.

Try it in Python

A short run jumps about. A long run settles near the true probability. This is why a bigger experiment is more convincing.

You should find that with 10 tries the band is very wide: a fair coin can easily give 2 or 8 heads. With 10000 tries it is only about 0.01 either side of the true probability.

Rolling a die

A fair die has a probability of $\dfrac{1}{6} \approx 0.167$ for each face. Roll it $600$ times.

The coin has no memory

A fair coin lands heads $5$ times in a row. Is tails now "due"? No. The next toss is a new toss, and the probability of heads is still $0.5$. The coin does not remember what happened before.

Practise with new numbers

Common mistakes

• Trusting a short run. Ten tosses can easily give 7 heads from a fair coin.
• Thinking a result is "due". A coin or a die has no memory.
• Expecting exactly the probability. Even a long run gives a share close to it, not exactly it.
• Calling a die unfair too soon. Some variation is natural.

Remember these

• Relative frequency = successes divided by tries. It estimates the probability.
• A short run jumps about. A long run settles near the true probability.
• A coin or a die has no memory.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A pin lands point up $130$ times in $200$ drops. How many times do you expect point up in $1000$ drops?

2. A fair coin lands heads $6$ times in a row. What is the probability of heads next?

3. Which gives a better estimate of a probability: 30 heads in 50 tosses, or 520 heads in 1000 tosses?

4. A die shows 95, 104, 99, 102, 97 and 103 of each face in 600 rolls. Is it unfair?