Eight to Late

It can be hard to develop an intutitive feel for a probability  that is expressed in terms of a single number. The main reason for this is that a numerical probability, without anything to compare it to, may not convey a sense of how likely (or unlikely) an event is. For example, the NSW Road Transport Authority tells us that  0.97% of the registered vehicles on the road in NSW in 2008 were involved in at least one accident.  Based on this, the probability that a randomly chosen vehicle  will be involved in an accident over a period of one year is 0.0097. Although this number suggests the risk is small, it begs the question: how small? How does it compare to the probability of other, known events?   In a short paper entitled, The Improbability Scale,  David Ritchie outlines how to make this comparison in an inituitively appealing way.

Ritchie defines the Improbability Scale, , as:

where is the probability of the event.

By definition, is 0 for absolutely certain events (), and increases as decreases. The advantage of using (as opposed to ) is that, in most case, , will be a number between 0 and 10.  An of 10 corresponds to a probability of 0.0000000001, which is so small that the event it refers to is practically impossible.

Let’s look at the improbability of some events expressed in terms of  .

1. Rolling a six on the throw of a die. = 1/6;  = 0.8.
2. Picking a specific card (say the 10 of diamonds) from a pack (wildcards excluded). = 1/52;  = 1.7.
3. A (particular) vehicle being involved in at least one accident in the Australian state of NSW over a period of one year (the example quoted in the in the first paragraph). = .0097;  =  2.0.
4. One’s birthday occurring on a randomly picked day of the year. = 1/365;  =  2.6.
5. Probability of getting 10 heads in 10 consecutive coin tosses. =   (or 0.00098 ); = 3
6. Drawing 5 sequential cards of the same suit from a complete deck (a straight flush). = 0.0000139;  =  4.9 (Note: This can be calculated by dividing the total number of sequential 5 card hands and dividing it by the total number of 5 card hands from a deck of 52. I’m too lazy to do the calculation myself, but it’s explained in this Wikipedia article if you’re interested. )
7. Being struck by lightning in Australia. = 1/1600000; =  6.2. (source: this article from Australian Geographic – the article doesn’t say over what period, but I reckon it’s per year)
8. Winning the Oz Lotto Jackpot. =  ; =  7.7 (based on odds from NSW lotteries for a single game)

Apart from clarifying the risk of a traffic accident, this tells me  (quite unambiguously!)  that I must stop buying lottery tickets.

A side benefit of the improbability scale is that it eases the tasks of calculating the probability of combined events. If two events are independent, the probability that they will occur together is given by the product of their individual probabilities of occurrence. Since the logarithm of a product of two number equals the sum of the numbers, for the combined event is obtained by adding their individual values. So the   for throwing a six and drawing a specific card from a deck is 2.5 (that is, 0.8+1.7), making it more unlikely than being involved in a vehicle accident. That certainly puts both probabilities in perspective.

In short: the improbability scale offers a nice way to understand the likelihood of an event occuring  in comparison to other events. In particular, the examples discussed above show how it can be used to illustrate and communicate the likelihood of  risks in a vivid and intuitive  manner.