Several studies, e.g this one, show that most people, including professionals in domains where knowledge of chance and odds matter, have difficulties understanding probabilities.
Take the following example (from the linked study above):
– it is known that in a population of women of a certain age group, the frequency of breast cancer is 1%.
– if a woman has breast cancer, the mammogram test procedure has a 90% probability of positive indication (i.e. revealing occurence of existing cancer cells, “true positive”)
– if a woman has not breast cancer, the probability of the test procedure is nevertheless 9 % to indicate cancer (“false positive”)
The question is then: given a woman has received a positive test (i.e indication of cancer) what is the probability of her in fact having cancer ?
Take a guess yourself: given that you have just received a positive test indicating occurance of cancer, and knowing that the test is 90% accurate in pinpointing actual occurence of cancer, what’s the probability that you indeed have cancer ?
80% of the doctors of the above study got it wrong. In most cases, severely wrong. I’ll later post the correct answer in a comment to this post.
This type of problems can be solved by Bayes Theorem, which in its simplest form can be stated as:
xy / (xy + z (1-x))
where x denotes the “prior” (in the example above, the statistical 1% cancer frequency in the population), y denotes the sought probability occuring given a conditional event (in the example, the positive indication) and z denotes the probability of the conditional event not impacting the result (“false positive”).
For the fun of it, let’s apply Bayensan resoning to an other problem I’ve written about lately, the current chaos in the public transportation system of Northern Stockholm:
It might be interesting to calculate the “survival odds” for the current operator, Arriva, i.e the probability that they will be able to continue running the traffic in Northern Stockholm, in light of both the severe financial pressures resulting from their winning the contract with a bid 30% lower than competition, and the current chaotic situation where traffic is more or less crippled, and customer satisfaction going down the toilet.
So, let’s apply Baynesian reasoning to this problem:
Let’s say that the “prior” probability for Arriva to give up is 25%, that is, due to the severe financial pressures of their low bid, we estimate their ability to successfully manage their business in Northern Stockholm to 75%
Let’s further say that should Stockholm Traffic Council (SL) decide that Arriva has failed so badly in its operations that they decide to trigger the financial penalty clauses of the contract, then the probability of Arriva’s survival goes down to 50%
Finally, lets assess that the probability of Arriva staying in business given being hit by financial penalty is 10%.
Plugging these numbers into Bayes Theorem gives the posterior probability, i.e. the probability of Arriva staying in business after having been hit by penalties, at 37%.
So, should Arriva face financial penalty clauses, the probability of them leaving Northern Stockholm is almost 2/3. Perhaps it might be a good idea to short some Arriva stock… 🙂
Obviously, the result above is fully dependent on the input numbers, but the point of this exercise is not to provide an accurate probability prediction for Arriva defaulting in Northern Stockholm, but to demonstrate how Bayes Theorem can be used.