The purpose of this unit is to integrate student learning in the mathematics and social sciences learning areas.

Students develop their skills and knowledge on the mathematics learning progressions; interpreting statistical and chance situations, in the context of flooding as an example of natural catastrophes.

Students will solve theoretical and model practical probability problems in the context of natural disasters. The students will be encouraged to consider a probability as a proportion of specified outcomes to the total number of outcomes in a probability distributions and to express this proportion as a percentage, fraction or a decimal. Students will design and carry out a probability simulation to find the experimental probability of a probability situation that is more complex than the theoretical probability calculations of this level of study, ie situations that involve two or more conditional probabilities.

**Introductory session**

(This activity is intended to motivate students towards the context/integrated learning area and to inform teachers of the location of their students on the learning progressions):

The severity of a natural catastrophe (eg a one in 10 year or one in 100 year event, etc) is defined by the number of years that would be expected to elapse between two such events. A 1 in 10 year flood is expected to occur once within any given ten year period. The probability of a 1 in 10 year flood occurring in any given year is 10%. This means that on any given year there is a 10% chance of such a flood occurring. It is not likely, but it is possible for two such floods to occur in a given year. What is the probability of two such floods, 1 in 10 year floods occurring in the first two years of a ten year period?

In this activity, the teacher(s) will be able to locate their students on the interpreting statistical and chance situations learning progression by observing how students solve a problem involving combined probabilities.

The activity itself has a geographical focus, with a knowledge of statistical and probability processes needed to discuss the likelihood of various chance situations occurring. Mathematical discussion that should follow this activity involve:

- Chance situations and the likelihood of various outcomes.
- How a probability may be used to predict what might, but may not be used to determine what will, happen.
- The different formats of probability (fraction, decimal, percentage, ratio).
- How probabilities combine for independent events.
- Theoretical and experimental probabilities; differences, practicalities, etc.

**Session two**

Focussing on combining probabilities to find the probability of an outcome from a more complex chance situation.

**Focus Activity **

The probability of a 1 in 10 year flood occurring in any given year is 10%. This means that on any given year there is a 10% chance of such a flood occurring. It is not likely, but it is possible for two such floods to occur in a given year. What is the probability of two such floods, 1 in 10 year floods occurring in two consecutive years?

Discussion arising from activity:

How could we calculate this event occurring? How would our answer change if the the question was two such floods in three years?

**Building ideas**

We will work through how to find the probability of two 1 in 10 year floods occurring in three consecutive years?

- What is the probability of a 1 in 10 year flood occurring in any year?
- What is the probability of a 1 in 10 year flood not occurring in any year?
- List the sample space for this probability distribution. This is a list of all the ways the floods could occur over a three year period. (For example, one way could be flood, flood, no flood)
- Apply the probabilities of the event of flooding or not for each arrangement of two floods over the three years. (For example flood, flood, no flood would be 1/10 of 1/10 of 9/10, which is 0.1 x 0.1 x 0.9)
- Find the probability of two 1 in 10 year floods occurring in three consecutive years. (This is the sum of the probabilities found for the last question.)

**Reinforcing ideas**

- What is the probability of a 1 in 5 year flood occurring in any given year?
- Find the probability of one 1 in 5 year floods occurring in five consecutive years.
- Find the probability of two 1 in 5 year floods occurring in five consecutive years.

**Extending ideas**

- What is the probability of a 1 in 20 year flood occurring in any given year?
- Find the probability of one 1 in 20 year floods occurring in five consecutive years.
- Find the probability of two 1 in 20 year floods occurring in three consecutive years.
- What is the probability of a 1 in n-year flood occurring in any given year?
- Find the probability of one 1 in n-year floods occurring in three consecutive years.

### Session three

Focussing on running a simulation to find an experimental probability.

**Focus Activity **

- The probability of a 1 in 10 year flood occurring in any given year is 0.1
- The probability 1 in 10 year floods occurring in the three out of ten consecutive years is 0.0574. How could this probability have been found?

Discussion arising from activity:

Introduce the idea of probability simulations, using random numbers and using the theoretical probability for a flood to occur in any given year. How can the experimental probability be as accurate as possible (increase the number of trials – 50 is a reasonable number).

**Building ideas**

Use a die to run a probability simulation of 1 in 6 year floods occurring in the three out of ten consecutive years. Run at least 30 trials to find the experimental probability of your simulation.

**Reinforcing ideas**

Use random numbers to run a probability simulation of 1 in 10 year floods occurring in the three out of ten consecutive years. Run at least 30 trials to find the experimental probability of your simulation.

**Extending ideas**

Use random numbers to run a probability simulation of 1 in 20 year floods occurring in the three out of ten consecutive years. Run at least 30 trials to find the experimental probability of your simulation.

### Session four

Focusing on understanding the differences between theoretical and experimental probabilities, and discussing the advantages of each.

**Focus Activity**

- A meteorologist ran a storm model to give an experimental probability that an approaching storm will result in a major flood of 0.324
- A statistician calculated the theoretical probability of the same storm resulting in a major flood as being 0.2978
- Comment on the similarities and/or difference of these two probabilities for the same event. Can they both be correct?

Discussion arising from activity:

- These numbers are similar – with what precision (number of d.p.) can we say the probability of the approaching storm causing a major flood would be?
- What are the advantages and disadvantages of each method of gaining a probability?
- What would the effect of increasing the number of trials in the meteorologist’s simulation have on the agreement between the two probabilities.

**Building ideas**

Consider this situation. The probability of a southerly storm event next weekend is 50%. One sixth of all storms of this type result in flooding.

- Find the experimental probability that there will be a flood next weekend. Run at least 30 trials in your simulation.
- Calculate the theoretical probability that there will be a flood next weekend.
- Compare your experimental probability with the theoretical probability, that you calculated for the same event

**Reinforcing ideas**

Consider this situation. The probability of a storm event next Wednesday is 25%. One tenth of all storms in recent history have resulted in flooding. One third of all floods are likely to be severe.

- Find the experimental probability that there will be a severe flood next Wednesday. Run at least 30 trials in your simulation.
- Calculate the theoretical probability that there will be a severe flood next Wednesday.
- Compare the two probabilities.

** Extending ideas**

Comment on the similarities and/or differences of the theoretical and experimental probabilities for whether there will be a severe flood next Wednesday. What do you expect the results of the simulation to be if you ran 50, or 100 trials?

### Session five

Focusing on finding a theoretical probability and expressing it as a percentage.

**Focus Activity**

A 1 in 10 year flood is expected to occur once within any given ten year period. The probability of a 1 in 10 year flood occurring in any given year is 10%.

What is the probability of a 1 in 100 year flood occurring in any given year? (note, since the probability given is in %, so to should your solution).

Discussion arising from activity:

What proportion is a year of the total time period specified in a 1 in 10 year event? In a 1 in 100 year event? How does this proportion relate to a probability expressed as a percentage?

**Building ideas**

Years between events | Probability of event in a given year | Percentage probability of event in a given year |

200 | ||

100 | 1/100 | 1 % |

50 | 1/50 | |

20 | ||

10 | 1/10 | 0.1 % |

5 | ||

2 |

**Reinforcing ideas**

- What is the percentage probability of a 1 in 200 year flood occurring in any given year?
- What is the percentage probability of a 1 in 1000 year flood occurring in any given year?
- What is the percentage probability of a 1 in 50 year flood occurring in any given year?

**Extending ideas**

Give a general rule for the percentage probability of a 1 in n year flood occurring in any given year.

### Session six

Focussing on showing understanding of dependence and independence in chance situations.

**Focus Activity**

If a farm experiences two 1 in 100 year floods in a single decade, should the owners rush out and buy a lottery ticket? Discuss your reasoning in terms of chance situations.

Discussion arising from activity:

Are the events of the two floods independent? (This is a debatable topic given the factors involved in storm systems and the trends of global warming). What about the events, a flood and a lottery? Will the probability of a win on the lottery be affected by the weather? Are they independent events?

**Building ideas**

A local raffle has 100 tickets. The probability of sunshine on the day the raffle is drawn is ½.

- What is the probability of buying one ticket and winning the raffle?
- What is the probability of buying the winning ticket and it being a sunny day?
- What is the probability the winning ticket and/ or being sunshine on that day?

**Reinforcing ideas**

The local lottery has 10,000 tickets. The probability of rain on the day the lottery is drawn is 20%.

- If all the lottery tickets are sold, what is the probability of winning the lottery on one ticket?
- What is the probability of winning the lottery on one ticket on a rainy day?
- What is the probability of having a ticket but not winning the lottery and it not raining on the day of the draw?
- What is the probability of either a win on a lottery ticket or rain on the day of the draw?

**Extending ideas**

Look at the sum of your answers to the last three questions. What do you notice? Could you use that knowledge to find another way of answering the last question?