# Episode 2

Reasoning Resources: Coin data from Worksheet 1, Worksheet 2 on OHT or A3/A2 sized, Worksheet 3 on OHT or A3/A2 sized
Whole class work
If we make the sample large enough, then the observed probability eventually approaches the causal probability (or in statistical terms, approaches the probability as estimated from analysis of the probability space). The point here for the pupils is how large the number of observations has to be: well in excess of 200, and for safety closer to S00 or 600. In this way we prefigure the notion of estimation — to be developed much later in instructional lessons. Draw a cumulative table of heads on the board.
The activity could be bridged to the previous work on probability. For example, in TM16: Three dice, how much variation was there between the numbers in the samples you took? The technical name for this is sampling variation. An earlier activity, TM 5 on length of words, also considered sample size. Fill in the results for each batch of 10 throws in the first column. Ask pairs with extreme results in one direction, e.g. nearer to 30%, to write in their results first, in order to highlight the expected convergence of the cumulative results towards 50% with the increase of sample size. You may want to indicate who each sample of 50 belongs to.

As the class fill in the cumulative column for total number of heads so far work out the percentage, using a calculator. Then plot the values from the table on to the cumulative chart (Worksheet 2) illustrating the one-to-one relationship between the table and the chart. (You may want to emphasise the bigger variations in the samples of 10 by plotting on the graph the proportions in those samples for the first two pairs of pupils.) Involve the class in checking how the proportion is behaving: Is it getting nearer or further away from the hypothetical 50%? Repeat until all the results are plotted.

• How do you know that the proportion of heads is really getting nearer to half and not just continuously fluctuating?
• What sample size is enough to convince us that the fluctuation is getting smaller?
• What are the variables in the graph? (sample size and cumulative proportion of heads).
• Mathematicians say that the limit value of the proportion is exactly 0.5. What do they mean? Mention the terms ‘frequency; relative’ and ‘limit.'
• Reflection on the best sample size
A fruitful discussion may follow a ‘second go; using the samples of 50 throws in a different order. Suppose that your first samples accidentally are all lower than the half-way value. Then the point where you reach reasonable closeness to the half-way mark is much later! In the case of a coin we can see that the sides are symmetrical in weight, and so we expect the number of heads and tails to be equal, if the coin is a fair one. But suppose we are in a different situation, with the results from some small samples, and we want to use the results to find out what the chances really are of the two possible outcomes.

Introduce a real-life context. We know that a baby can be a boy or a girl. But are there some causes after a disaster, or peculiar to the area or a historical period, which may give a bias favouring more boys than girls or vice versa? How many babies would we need to count in a hospital's records before we decided whether or not there is such a bias?

We can investigate by analysing data in each batch of 10 separately, Show the graph for samples of 10 on Worksheet 3. Can you see a pattern?

Alternatively we can add the batches to make a much larger sample, where the ratio is repeatedly recalculated as we did for the coin tosses. Show the graph for samples of 50 on Worksheet 3.

Discuss how large a sample of births you would need to collect from a hospital before you could decide if there was a bias in favour of boys. Pupils should see, in some way, that the outcomes for something like 500 or more births are going to be needed.