In “Occam's Razor” episode Gregory House MD substantiating his view that the hypothesis explaining the observed symptoms with two diseases is simplier, than a hypothesis based on one but very rare disease, speaks the following statement:
Each one of these conditions is about a thousand to one shot. That means that any two of them happening at the same time is a million to one shot. Chase says that cardiac infection is a 10 million to one shot, which makes my idea 10 times better than yours.
This statement shows that Gregory House is able to multiply two numbers and compare the result with a third one, he also heard something about probability and would like very much to use mathematics, but in spite of his sincere desire he cannot. (Generally, all of this can be probably said about the screenwriter who wrote the above sentence.)
First, House makes the assumption of independence of developing the two diseases, which are suspected in a patient. That does not have to be. For example, without any statistics, I can confidently say that the incidence of Kaposi's sarcoma and toxoplasmosis of the brain are not independent events. House as a doctor should be aware of the possibility of such dependence. Perhaps in the case of specific diseases he suspects, he would defend his assumption about their independence but in episode he did not mention even a word about it, though he certainly should.
Secondly, even if these diseases are independent, we are not considering here the likelihood of developing these diseases with random individuals about which we know nothing. Here we have a specific patient with specific symptoms. So we are dealing with the conditional probability not the ordinary one. Generally probability is used to estimate our ignorance about the specific events if they happen or not. Information we gain about the occurrence of such events as the symptoms change our knowledge and thereby the likelihood we assign incidents. Some symptoms of the disease are very specific and do not give a large area to doubt. Presence of Kaposi's sarcoma mentioned above is the unique basis for the diagnosis of end-stage HIV infection i.e. AIDS. Flu-like symptoms may also result from such a virus infection but may also be due to influenza therefore there is no such unambiguity as in the previous case. Symptoms did not influence on the probability of HIV infection among total population of humans, but they change our assessment of the most likely diagnosis for this specific case. House completely ignored this in his statement.
Let us forget for a moment House's mistake and accept that the incidence of diseases which he suspects are independent events and conditional probabilities with symptoms observed are proportional to those that mentioned the House. Still House's reasoning is erroneous. In the manner described conditional probability of occurrence of both diseases at once cannot be obtained without additional assumptions as a product of the corresponding conditional probabilities. The following examples illustrate this.
We independently draw two real random numbers from 0 to 1. Let the probability distribution in this experiment be uniform. Let the events A and B will be situations in which respectively the first and second number is less than 0.5. Let S be the event a situation in which the sum of two numbers is greater than the 1. Then, of course,
P(A ∩ B) = P(A) · P(B)
P (A | S) = P (A ∩ S) / P (S) = (1 / 8) / (1 / 2) = 1 / 4
P (B | S) = P (B ∩ S) / P (S) = (1 / 8) / (1 / 2) = 1 / 4
Exact numbers, moreover, are not meaningful. It is important that these values are nonzero and in the same time:
P (A ∩ B | S) = P (A ∩ B ∩ S) / P (S) = 0 / (1 / 2) = 0
Such a situation would mean a total defeat for the House and his reasoning.
But let us again forget about House's mistakes and assume that conditional probability of occurrence of of both diseases at once he suspects is actually ten times larger than the conditional probability for the disease, which presumed his interlocutor. Is it enough to assume that the procedure proposed by him is ten times better? Not really. Probability is not enough. It is necessary to further assess the risk of taking or not taking certain actions. Let us return to the example cited earlier. If you are experiencing flu-like symptoms that may result from influenza or HIV infection, you will make a HIV-test, despite the fact that the flu is more likely. Decisive for the decision of such actions is the risk to which we are exposed to in case of not making the test. In finance mathematics it is easier because the risk can be expressed quantitatively. The doctor must undertake a quantitative assessment of more difficult to capture factors such as death, disability or chronic pain. In the considered case House ignored it or at least said nothing about such considerations.
The mathematical ignorance of TV series hero would not be something specially terrible if the quoted passage has not become a model reasoning in his book “House and philosophy — everybody lies”. This example is used to illustrate that the simplicity criterion in selection of hypothesis is not always obvious thing. Indeed it is not, but what a pity that the authors of this book do not devote even a few critical sentences to this particular citation.