How can Bayes' theorem be of help in our life

Bayes' theorem relates the conditional and marginal probabilities of two random events:

P(A|B)=P(B|A) * P(A) / P(B).

This simple equation can be of help decoding some so called information provided by mass media. Consider this example.

From the “Morning News” of Tuesday, Feb 29th 2007:

20 out 100 foreign prisoners are from Malucistan.

Penitentiary authorities informed that among the foreign prisoners (a 37% of the total), 2 out of 10 are citizens of Malucistan. In our Country, lives some 350000 malucistanians. These figures produced some social alarm among the population, especially in those areas with a higher concentration of malucistanians.

The question is: is the social alarm justified? The value that should really interest the citizen should be “If I meet a malucistanian, what is the probability that he is somehow a dangerous/dishonest person, who deserves prison?” Let’s call this value of interest P(P|M).

Let’s summarise the information we’re given. We are said that 20% of foreign prisoners are malucistanian [let’s call this value P(M|F)] and that among the whole prisoners, a 37% are foreigners [let’s call this value P(F|P)]. Now, knowing the total population in the Country [say some ~60M] we can easily calculate the percentage of malucistanian [P(M)]. Finally, with a simple internet search we can obtain the percentage of prisoner in the Country [say, P(P)~=0.08].

We now have all the elements to answer our question.

In fact, thanks to Bayes,

P(P|M)=P(M|P) * P(P) / P(M)

and being P(M|P)=P(M|F) * P(F|P)

we readily obtain that when meeting a malucistanian, the probability that he deserves to go to prison is around 1%, that is, 1 out of 100, which sounds quite different from the 2 out of 10 highlighted by mass media as meaningful figure.

Moral: be cautious with mass media and always investigate a bit by yourself …

Note: Of course Malucistan and the "Morning News" do not exist. Nonetheless, the figures used for this small exercise closely reflect the real situation of a real Country and of a real foreign community leaving there.