# 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**.

### 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 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?)

## 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?**

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 | 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?**

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 | 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**, but 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?

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

## 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 |