# Random, or Not?

Before you begin this activity, you might lke to read these definitions:

• Whole numbers
Whole numbers are the numbers {0, 1, 2, 3, ...} etc.
There is no fractional or decimal part. And no negatives.

• Random
Random means: Without order. Not able to be predicted. Happening by chance.

• "Equally likely" means that each possible outcome from an experiment has the same chance of occurring (example: when you toss a fair die, each of the six faces is equally likely to land face up).

## Add or Multiply Two Whole Numbers Together

Have you ever thought about what result you get:

• When you add two whole numbers together?
• Or when you multiply two whole numbers together?

### In particular, are all last digits equally likely?

Example:

39 + 57 = 96 has last digit 6

38 × 45 = 1,710 has last digit 0.

### Adding. Tick one of the following:

 When you add two randomly selected whole numbers Tick Yes, last digits are all equally likely No, last digits are not all equally likely

### Multipling. Tick one of the following:

 When you multiply two randomly selected whole numbers Tick Yes, last digits are all equally likely No, last digits are not all equally likely

### Let's see whether you guessed correctly ...

(Note: we give answers to the tables at the bottom of the page ... but only check them when you are done, or this wouldn't be an activity would it?)

• 13 + 18 = 31,
• 23 + 78 = 101,
• 53 + 68 = 121, and
• 83 + 58 = 141

You will see that they all end in the digit 1.

### So what do they have in common?

They are all sums of whole numbers whose last digits are 3 and 8 respectively. When we add a number ending in 3 to a number ending in 8, we always get a number ending in 1.

So all we need to consider are the last digits of the two numbers we are adding together.

### We can do this by completing a table.

The following table is incomplete. Can you fill in the missing numbers?

Remember: just the last digit after addition, so with 6+7=13, we want the "3"

+ 0 1 2 3 4 5 6 7 8 9
0 0 2 3 5 8
1 2 4 7 0
2 2 5 7 0 1
3 4 7 8 0 1
4 4 6 8 0 3
5 6 8 0 1 3
6 6 8 0 2 3
7 8 0 1 3 6
8 8 0 3 5 6
9 0 1 3 6 8

You can now tally the numbers and complete a frequency table:

 Last digit Tally Frequency Relative frequency 0 10 0.1 1 2 3 4 5 6 7 8 9

### Did you find all last digits are equally likely this time?

Each value 0 to 9 occurs exactly 10 times out of 100.
So they are all equally likely, just like when you throw a die.

### Relative frequencies

Can you complete the last column of the table with the relative frequencies for each last digit?

## Multiplication

• 12 × 19 = 228,
• 22 × 79 = 1,738,
• 52 × 49 = 2,548 and
• 82 × 39 = 3,198

You will see that they all end in the digit 8.

### So what do they have in common?

They are all products of whole numbers whose last digits are 2 and 9 respectively. When we muliply a number ending in 2 with a number ending in 9, we always get a number ending in 8.

So all we need to consider are the last digits of the two numbers we are multiplying together.

The following table is incomplete. Can you fill in the missing numbers?

Remember: just the last digit after multiplication, so with 3×6=18, we want the "8".

× 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 3 4 6 7 9
2 0 4 6 0 4 6
3 0 3 9 2 8 1 7
4 0 8 2 0 4 2 6
5 0 5 0 0 0 5 0 5
6 0 2 8 0 6 8
7 0 7 1 8 2 9 3
8 0 6 4 0 8 4 2
9 0 9 6 4 3 1

You can now tally the numbers and complete a frequency table:

 Last digit Tally Frequency Relative frequency 0 27 0.27 1 2 3 4 5 6 7 8 9

### Did you find all last digits are equally likely this time?

Last digit 0 occurs 27 times out of 100, but last digit 7 only occurs four times:

1 × 7, 3 × 9, 7 × 1 and 9 × 3

### Relative frequencies

Can you complete the last column of the table with the relative frequencies for each last digit?

### Example

0 occurs 27 times out of 100, so the relative frequency for 0 is 27/100 = 0.27

## Conclusions

Did you predict the results correctly?

Addition gives equally likely results but multiplication does not ... how about that!

Could the relative frequences be useful somehow?

... Don't look past here until you have completed the activity! ...

## Completed Tables

+ 0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 0
2 2 3 4 5 6 7 8 9 0 1
3 3 4 5 6 7 8 9 0 1 2
4 4 5 6 7 8 9 0 1 2 3
5 5 6 7 8 9 0 1 2 3 4
6 6 7 8 9 0 1 2 3 4 5
7 7 8 9 0 1 2 3 4 5 6
8 8 9 0 1 2 3 4 5 6 7
9 9 0 1 2 3 4 5 6 7 8

 Last digit Frequency Relative frequency 0 10 0.1 1 10 0.1 2 10 0.1 3 10 0.1 4 10 0.1 5 10 0.1 6 10 0.1 7 10 0.1 8 10 0.1 9 10 0.1 Total 100 1.0

### Multiplication

× 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 0 2 4 6 8
3 0 3 6 9 2 5 8 1 4 7
4 0 4 8 2 6 0 4 8 2 6
5 0 5 0 5 0 5 0 5 0 5
6 0 6 2 8 4 0 6 2 8 4
7 0 7 4 1 8 5 2 9 6 3
8 0 8 6 4 2 0 8 6 4 2
9 0 9 8 7 6 5 4 3 2 1

 Last digit Frequency Relative frequency 0 27 0.27 1 4 0.04 2 12 0.12 3 4 0.04 4 12 0.12 5 9 0.09 6 12 0.12 7 4 0.04 8 12 0.12 9 4 0.04 Total 100 1.00