The bitter-sweet taste of toxic substances
Box 1 | Chances and risks
Every decision involves weighing up risks and benefits. We make some decisions so naturally that we often don’t even notice we’re doing it. Other decisions aren’t made so easily and require us to be well-informed about what the risks and benefits are and the likelihood of them occurring.
Mathematics of chance – probability
To calculate a risk numerically uses probability – the mathematics of chance. The calculation of probability is simple. You first count the number of ways something can happen and then divide this number by the total number of things that can happen.
An example of throwing a dice can illustrate how probability is calculated. When you throw a dice, it lands showing one of six numbers – there are six possible outcomes, and only one way each of them can happen. So the chance of throwing a ‘five’, for example, is 1 in 6 (one sixth or about 0.17). Probabilities are often expressed as percentages which describe how many times an event happens out of 100 times. One sixth is about 17 per cent, so if you tossed a dice 100 times, you would expect to roll a ‘five’ about 17 times.
Weighing up pluses and minuses
One of the simplest ways to weigh up risks and benefits is to add up the consequences multiplied by the probability that they will happen. This gives you an overall expected outcome, as shown in the following example.
A lucky dip has 100 tickets, each of which costs $5. The prizes are allocated to the tickets as follows:
- one is a prize of $100;
- one is a prize of $40;
- two have prizes of $5;
- 95 tickets have no result; and
- one is an unlucky ticket (you have to forfeit $30).
There is a 1 in 100 chance of choosing a particular ticket. The expected outcome is:
$100×1/100 + $40×1/100 + $5×2/100 + $0×95/100 – $30×1/100 = $1.20
This means that if you played many new games of lucky dip, you would sometimes gain and sometimes lose, with an average gain of $1.20 each time. (You could never actually win $1.20, that’s just an average.)
But the tickets cost $5 each, so the expectation is that you will lose an average of $3.80 each time you play. Games of chance are designed to enrich the person running the game!
The probability of more than one event happening
You can calculate the chances of more than one independent event happening by multiplying together the chances of each one happening. For example, there’s a party on Saturday night and the probability that your best friend can go is 0.6 (and that they can’t go is 0.4). The probability of your worst enemy going is 0.3 (and not going is 0.7).
The probability that both will go is 0.6 × 0.3 = 0.18.
The probability that neither will go is 0.4 × 0.7 = 0.28.
The probability that just your enemy will go is 0.4 × 0.3 = 0.12.
The probability that just your friend will go is 0.6 × 0.7= 0.42.
(Notice that all these probabilities add up to 1 because they cover every possibility.)
We have assumed that the two ‘events’ (your friend or your enemy attending) are ‘independent’. The calculation will be different if your friend and your enemy discuss the party by telephone before deciding whether or not to attend!
While you may not use numbers, your brain usually does a quick calculation before making decisions (eg, when deciding how much to spend on a raffle ticket or whether to go to a party or not.) Although these are trivial examples, similar kinds of calculations about chances and risk can be used to help make more important decisions (eg, those involving health and safety).
Risk perception
Many factors influence how people perceive risks and make decisions. More people worry about travelling by plane than by car, even though statistics show that they are 10 times more likely to be killed in a car travelling between two cities than in a commercial airline flying the same journey. More accidents happen in kitchens than anywhere else, but you don’t think twice about walking into your kitchen. Our perception of risk is often based on how dreadful we think a risk is and how well we understand the risk, rather than on probabilities.
Using mathematics to analyse risks and benefits and to analyse the probability of an event occurring can help us make decisions. But we still have to decide how to measure the benefits and risks, and how best to weigh up the options.
Boxes
Box 2. Cyanide and arsenic
Box 3. DDT and biological concentration
Related site
Scientific responsibility and the public perception of risk (Transcript from ABC radio's Ockham's Razor, 2 December 2001)
Further reading
Risky business (by Ruben Meerman, The Helix, No. 63, December 1998/January 1999, pages 28-29)
The lore and lure of dice (by Ian Stewart, Scientific American, November 1997, pages 76-78)
Posted February 1999.