On The Ball Statistics In Your World 
Student Notes
Teachers Notes
Picking Results
 
Settling Down
 

What Are the Chances?

Picking Results
To win a big prize on the pools you must guess which games will be score draws next Saturday. Some people study past results. Others use a pin or guess. These are not truly random methods.

a Which method do you think is best?
b Turn to Table 1. Close your eyes and make a dot with a pencil on these results. Write down the result nearest to your dot. Give your result to your teacher who will give you the class results.
c What fraction of your class picked a home win?
What fraction picked an away win? What fraction picked a score draw?

In Table 1 there are 46 matches: 22 home wins, 11 away wins, 10 score draws and 3 no-score draws.

Fraction of away wins = 11/46 = 0.24 (to 2 sig.figs)

The chance that a match picked at random is an away win is 0.24. Another word for chance is probability. We write, using Table 1 results:

Probability (picking an away win) = 0.24

d Using Table 1 results, write down, to 2 sig. figs:
Probability (picking a home win)
Probability (picking a score draw)
Probability (picking a no-score draw)
Probability (picking a draw)

 

Settling Down
Since the games in Table 1 have already happened we know the probability of a draw if one of these games is picked at random.

Usually, however, we want to predict draws for next week. One Saturday's results give some idea of what might happen next Saturday. But results are not the same every Saturday. Some weeks there are few draws and big prizes on the Pools. Other weeks there are many draws. It is better to collect more results because the proportions settle down. We use these proportions to estimate the probabilities for future games.

In Table 4, home wins over the first few weeks of the 19751/76 season are given. The first line shows the number of home wins in the 46 matches played on August 22, 1975. The second line shows the number of home wins in the 90 matches plaved on August 22, 1975 and August 29, 1975, the first two Saturdays of the season. Each of the next four lines then adds in the results of another week. The last line shows the number of home wins and matches played in the whole season.

All Saturday matches played before Home wins Matches Fraction: home wins/matches
Aug 22, 1975 19 46 19/46=0.41
Aug 29, 1975 41 90  
Sept 4, 1975 65 131  
Sept 11, 1975 85 173  
Sept 18, 1975 109 216  
Sept 25, 1975 126 258 126/258=0.49
All season 1019 2028  

Table 4 - Home wins 1975-6

a Use a calculator to help you complete the last column to 2 significant figures. Write the answers in your book.
b As more results are included, what happens to the fraction of home wins?

There is a similar pattern for away wins and draws. Over the whole season, the proportion of home wins is about 0.5; for away wins it is about 0.2; for draws it is about 0.3. These figures give us an idea of what to expect in the future.

We say:

Probability (home win) 0.5
Probability (away win) 0.2
Probability (draw) = 0.3

Clearly, the home team has an advantage. Someone said that the away team should get both points even if the result is a draw.

* c Do you think that would be fair?
* d Can you think of any other way of removing the advantage to the home team?

 

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