# Random, or Not?

Before you begin this activity, you may need to familiarize yourself with some 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, would all **last digits** be equally likely?

Example:

**39 + 57 = 96** has last
digit **6**

**38 × 45 =
1,710** has last digit **0**.

### So are the digits 0 to 9 all equally likely?

What's your guess?

**Adding.** Tick one of
the following:

When you add two
randomly selected whole numbers |
Tick |

Yes, last digits are alll equally likely | |

No, last digits are not alll equally likely |

### **Multipling.** Tick one of
the following:

When you multiply two
randomly selected whole numbers |
Tick |

Yes, last digits are alll equally likely | |

No, last digits are not alll 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?)

## Addition

Think about:

- 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?**

+ | 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 |
Relativefrequency |

0 | 10 | 0.1 | |

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 |

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

**The answer is YES.
**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?**

### Example:

**0 **occurs **10** times out of **100**, so the relative frequency
for **0** is **10/100 = 0.1**

## Multiplication

Think about:

- 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?**

× | 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 |
Relativefrequency |

0 | 27 | 0.27 | |

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 |

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

**The answer is still NO.**

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

1 × 7,3 × 9,7 × 1and9 × 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?

## Completed Tables:

Here are the answers:

### Addition

+ | 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 |
Relativefrequency |

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 |
Relativefrequency |

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 |