# 3.1 Expressing probability

Probability can be expressed mathematically in a number of ways. The following all show an equal probability:

- as a proportion (e.g. 1 in 8)
- as a decimal number between 0 and 1 (e.g. 0.125)
- as a fraction (e.g. one divided by eight )
- as a percentage (e.g. 12.5%).

An impossible event has a probability of 0, while an event that is certain to happen has a probability of 1. For instance, the probability of dying (eventually) is 1.0 (or 100%) for everyone.

Figure 12 illustrated scales of risk reduction. This is revisited in Table.3, where probabilities (or risks) are expressed in the different standard forms, including scientific notation as a useful reminder.

Proportion | Decimal | Fraction | Percentage | Scientific notation |
---|---|---|---|---|

1 in 2 | 0.5 | 50% | 5 ×10^{−1} | |

1 in 5 | 0.2 | 20% | 2 ×10^{−1} | |

1 in 10 | 0.1 | 10% | 1 ×10^{−1} | |

1 in 100 | 0.01 | 1% | 1 ×10^{−2} | |

1 in 1000 | 0.001 | 0.1% | 1 ×10^{−3} | |

1 in 10 000 | 0.0001 | 0.01% | 1 ×10^{−4} | |

1 in 100 000 | 0.000 01 | 0.001% | 1 ×10^{−5} | |

1 in 1 000 000 | 0.000 001 | 0.0001% | 1 ×10^{−6} | |

1 in 10 000 000 | 0.000 000 1 | 0.000 01% | 1 ×10^{−7} |

Professional risk analysts often quote various figures showing the risks associated with everyday activities in comparison to risks in which they have an interest. Some examples from one published table of risks are included in Table 4. However, remember that all such tables and figures are based on historical information and cannot always be relied upon for future predictions.

Risk of dying in a year | Activity |
---|---|

1 in 100 | Five hours of solo rock climbing every weekend |

1 in 1000 | Working in a high-risk group within the more hazardous industries |

1 in 10 000 | General travelling by road or on foot |

1 in 100 000 | Working in the very safest parts of industry |

1 in 1 000 000 | Fire caused by a cooking appliance at home |

1 in 10 000 000 | Being hit by lightning |

One obvious reason why future risks cannot be predicted easily is that all professions are constantly working to improve safety. Therefore the two industrial death probabilities quoted above will now be lower. The number of car accidents, for example, has diminished rapidly in recent decades in the UK, and injury rates have also decreased with the mandatory use of safety belts and the installation of airbags, impact-protected body shells and better tyres. Improvements to road design and layouts have also played a part here. These improvements do not mean that we should be complacent; there are almost always additional ways in which death and injury rates can be reduced yet further.

## Activity 3 Risk in scientific notation

Complete the following table to express the risks shown in terms of a decimal, a fraction and scientific notation.

Decimal | Fraction | Scientific notation |
---|---|---|

0.05 | ||

1.6 ×10^{−2} | ||

### Answer

Decimal | Fraction | Scientific notation |
---|---|---|

0.05 | 5 ×10^{−2} | |

0.016 | or | 1.6 ×10^{−2} |

0.006 67 | 6.67 ×10^{−3} | |

0.000 000 333 | 3.33 ×10^{−7} |

The important thing is to understand that although there might be a ‘1 in 10 million’ risk of death by lightning, or a ‘1 in 100’ risk of death from five hours of solo rock climbing every weekend, these figures may not show an accurate measure of risk from that activity. There are many other factors that might change these estimates – for instance, an individual might be a very safety-conscious rock climber or might make a habit of standing under trees during lightning storms.