Being Fair To Ernie

Being Fair To Ernie	Statistics In Your World

Student Notes

Teachers Notes

A Raffle

Are They Random

Random Number Tables

Three Tickets

All the games you played gave a counter to one of three people. Sometimes we need a fair way to choose one person from more than three.

A Raffle
Ann is running a small raffle for 10 friends. She has 10 numbered tickets, numbered 0 to 9. Each friend buys one ticket. Ann has to find a fair way of choosing a number from 0-9 so that each number is equally likely.
She does not like Choosing Numbers. The numbers are not equally likely. It is not fair.
She does not like Names in a Bag. It is fair, but the pieces of paper are fiddly.
Here are some ways she tried. Try two of them to see if they are fair.

Method 1

You will need: A table like Table 2 and EITHER a 10-sideed spinner OR (to make your own spinner) two pieces of card, a drawing pin, a ruler, a protractor, and a pair of scissors.]

1	On one piece of card, draw a circle with radius 5 cm.
2	Draw a radius. Then mark off another radius at 36⁰ from the first. Continue marking radii at 36⁰ until you have 10. Mark the sectors 0 to 9.
3	Cut out the circle. Push the drawing pin through the center of the circle and into the other piece of card.
4	Draw an arrow on the second piece of card pointing to the center of the circle (see Figure 2).

Figure 2 - Home-made spinner

Here is how to work your home-made spinner. Hold the drawing pin with one hand. Flick the circle with your other hand, and read the number against the arrow.

Spin the spinner. When it stops, note the number resting on the table or against the arrow.

Record your results in a table like Table 2.

Table 2 and Figure 3 show Ann's results.

Number	Tally	Total
0		4
1		5
2		3
3		6
4		3
5		5
6		4
7		4
8		2
9		4

Table 2 - Anne's Results

Figure 3 - Bar chart of Ann's results

	a	Repeat until you have 4o results.
*	b	Plot these results on a bar chart like Figure 3.

Method 2

You will need a coin and a die.
Toss the coin and roll the die. If the die shows a six, roll it again.

Die

1 2 3 4 5 6

Tail

0 1 2 3 4 Throw

Tail

5 6 7 8 9 again

Table 3 - Rules for scoring with a coin and a die

Examples:

	d	Record your results in a table like Table 2.
	e	Repeat until you have 40 results.
*	f	Plot these results on a bar chart like Figure 3.

Method 3

You will need an icosahedral die.
(An icosahedral die has 20 faces with faces labelled 0-9 twice. Be careful not to wix up a 6 with a 9.)

	g	Record your results in a table like Table 2.
	h	Repeat until you have 40 results.
*	i	Plot these results on a bar chart like Figure 3.

Method 4

You will need a coin.
The coin is thrown four times. Record H for a Head and T for a Tail in the four columns of a table like Table 4 tells you which number to write down.
The first line of Table 5 shows you how to record, Tail, Head, Head, Head. Check with Table 5.

Third and fourth throws

First two throws	TT
	TH
	HT
	HH

0	1	2	3
4	5	6	7
8	9	10	11
12	13	14	15

Table 4 - Obtaining numbers with four coins

First two throws

Third and fourth throws

Number

Table 5 - Results of throwing a coin four times

	j	What number is given by line 3 of Table 5? Toss a coin four times.
	k	Record your results in a table like Table 5.
	l	Repeat until we have 20 numbers less than 10.
	m	Find a friend who has also used this method.
	n	Record yours and your friend's results in a table like Table 2.
*	o	Plot your combined results on a bar chart.

Fair numbers chosen in ways like these are called RANDOM NUMBERS. At each turn every number from 0 to 9 is equally likely to happen.

Which Method?
Look at the two methods you used for getting numbers.

Are they fair?

Are They Random?
Here are some ways of collecting figures. Say whether they would be RANDOM numbers. Give a reason for your answer.

	a	Read the last figure on the number plates of cars passing the school.
	b	Read the first figure on the number plates of cars passing the school.
	c	Read the last figure of telephone numbers from a page in the telephone directory.
	d	Write down a list of numbers 'at random' from your head.

Random Number Tables
There are several ways to produce random numbers. Each method takes a lot of time. A list of 40 random numbers using only one of the methods described above might look like

7, 7, 0, 4, 0, 1, 0, 9, 7, 3, 8, 9, 8, 4, 3, 5, 7, 5, 7, 6,

1, 2, 3, 9, 4, 3, 6, 4, 9, 7, 4, 0, 8, 3, 9, 9, 1, 8, 2, 6

Table 6 - A list of 40 random numbers

Sometimes we need a lot of random numbers quickly. A computer can give us some. It can print put random number tables.
Here is a random number table. There are no commas, and the numbers are printed in pairs. The list in Table 6 would be printed as:

77 04 01 09 73 89 84 35 75 76

12 39 43 64 97 40 83 99 18 26

Rules for Using Random Number Tables

1	Start anywhere. Close your eyes and touch the paper. When you open your eyes, write down the number nearest your finger.
2	Move your finger in any direction on the paper. Keep it moving in this direction.
3	Write down each figure you touch. Don't miss any out.

One-figure Random Numbers

	a	Use your random number table to write down 40 numbers.
	a	Record your results in a table like Table 2.

All your random numbers so far have been from 0 to 9. Sometimes we don't need all these. For example:

To find random numbers from 1 to 7, ignore 0,8, and 9 when they occur:
The list in Table 6 gives:

7, 7, 0, 4, 0, 1, 0, 9, 7, 3, 8, 9, 8, 4, 3, and so on.

Use the list in Table 6 to write down 10 random numbers.

	c	From 0 to 5
	d	From 2 to 6

Two-figure Random Numbers
Sometimes we need two-figure random numbers. For example: to get random numbers from 0 to 99, use the numbers in pairs.

The list in Table 6 gives:

77, 04, 01, 09, 73, 89, 84, 35, and so on.

that is: 77, 4, 1, 9, 73, 89, 84, 35

Use the random number tables to write down 10 numbers from 0 to 99.

Sometimes we don't need all these numbers. For example:
To get random numbers from 1 to 56, se the numbers in pairs.
Ignore 00 and all numbers from 57 to 99.

Table 6 gives:

77, 04, 01, 09, 73, 89, 84, 35, 75, 76, 12, 39, and so on.

that is: 4, 1, 9, 35, 12, 39

	f	Use the list in Table 6 to write down 10 random numbers from 1 to 74.
	g	Use your random number table to write down 10 random numbers from 1 to 32.

Three Tickets
Your teacher has three tickets for a pop concert. He wants to give them fairly to members of your class, so he is going to use random numbers.

	a	How could you number the members of your class? Do this.
	b	Is it fair to let any pupil have a chance of getting more than one ticket?
	c	What will you do if the same random number comes up twice?
	d	Use random number tables to find the numbers of the lucky pupils.
	d	What are their names?

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