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Introduction

In this assignment you will learn about calculating and expressing probabilities and then relating them to risk. These concepts will reoccur throughout this course, and so it is very important that you work not only to get correct answers but also to understand these problems so you can use these tools later.

Probability of a single event

Many natural disasters occur without patterns—they are random events, like a roll of a die or the flip of a coin. The likelihood of a particular event occurring, like flipping a

“heads” during a coin toss can be calculated. This is called the probability.

Probabilities can be expressed in a variety of ways, most often as a percentage, a fraction, or a decimal.

Example 1: What is the probability of rolling a 6 on a die?

Since there are 6 possible outcomes, and only one is the one we want, the probability is 1 in 6. This would be written in fraction form as 1 .

6

To find this in decimal form, divide the 1 by the 6 and you should get

0.1667.

To convert to a percentage, multiply by 100 to get 16.67%

Often we will need to determine the probability of an event not happening. Since the probability of an event occurring and the probability of an event NOT occurring must add up to 100% (or 1 in decimal and fraction form), the probability of an event not occurring is 100% minus the probability of the event occurring.

Example 2: What is the probability of NOT getting a 6 on the roll of a die?

Give in fraction, decimal, and percentage forms.

Since the probability of getting a 6 is 16.67%, then the probability of not getting the outcome must be:

100% — 16.67% = 83.33%.

This is also

5,

6

or 0.8333.

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1.10 Understanding the Math of Probability and Risk

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Recurrence intervals

Often hazards are communicated in terms of recurrence intervals. For instance, the

“100-year flood” really means a flood that has a 1% (1/100 or 0.01) of occurring in any year. When given a recurrence interval, the probability of the event is: where: P = 1/R

P = the probability in any given year

R = the recurrence interval

Example 3: What is the probability of a 50-year flood in any given year?

A 50-year flood means the recurrence interval R = 50. Using P = 1/R:

P = 1/50 or 2%

Probabilities for a series of events

We will now look at the funny way we calculate probabilities for a series of events. This is needed when you know the probability of an earthquake occurring in one year but you want to know what the probability is over some longer time period, say 30 years.

At first, you might think that the way to do this would be to add the probabilities, but this doesn’t quite work. This is shown by the fact that if you add the probabilities, then the probability of getting heads if you flip a coin twice would come out to be 100% since each flip would be have a 50% probability of occurring. But 100% means that it would be a sure thing, and we know that you won’t always get heads if you flip a coin twice. To correctly determine the probabilities of an event occurring when it occurs multiple times, we actually determine the probability of the event NOT occurring and then multiply those probabilities. This can be written as an equation:

1 – Ps=(1 – Pi)N where: Ps = the probability of the series of events (in decimal form)

Pi = the probability of an individual event (in decimal form)

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1.10 Understanding the Math of Probability and Risk

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N = the number of times the event could occur (ex: # of times a die is rolled.)

Example 4: What is the probability of getting “heads” if you flip a coin twice?

The probability of NOT tossing a “heads” in one flip is 50% and since you are doing it two times, the probability of you NOT tossing “heads” in two tries is

50% x 50% (or 0.5 x 0.5). That is 25%. So the probability of NOT tossing

“heads” in two flips is 25% - and the probability of…