Probabilities for two bombs and a businessman 
A businessman travels by air often. He computes the probability $p$
  for a flight to have a terrorist bomb on the plane, taking many
  factors into account. The businessman does not find $p$ small enough,
  and says to himself that under reasonable independence assumptions the
  probability to have two (unrelated) bombs on the same flight is $p^2$
  which is much smaller than $p$. So he makes the decision to secretly
  carry a (neutralized) bomb during each and every one of his flights,
  feeling much safer this way.

A probability exercise book I read some time ago asked for a refutation
of the businessman's fallacy in this situation. 
The only thing I can think of so far is that the choice of the probability model should
be made clearer. Any ideas for more convincing explanations ?
 A: It's wrong because he hasn't taken into account all the information he has. He calculates that with the knowledge he has at the beginning that the probability is $p$ for a single bomb and $p^2$ for a second independent bomb. These probabilities are the same but call then $p_1$ and $p_2$ for now so that the prob of two bombs is $p_1p_2$. He decides that by taking his own that he makes up one of those $p_i$'s, say his one is $p_2$, but he $\mathbf{knows}$ that he has taken a bomb on the plane and so with this extra information $p_2 = 1$. Hence the probability of two bombs on the plane (one of which is his) is $$p_1p_2 = p_1\cdot 1 = p_1$$
Hence we still have a $p_1 = p$ chance of another bomb on the plane. The important thing here is that probabilities are related to the amount of knowledge we have about the situation we are considering. 
A: The problem is the probability of his bomb being on the flight is 1 and is not $p$. Hence he can not claim the probability of two bombs being on the flight is $p^2$. There is no way for him to have his bomb occur with probability $p$ and also know that it is on the flight with him - its one or the other not both.
A: This is not an answer but it is too long for a comment.
This was a joke which started in the very early 70's  after the events described here.
A man comes to TWA headquarters in Saint-Louis, Missouri (at this time, TWA was one of the largest airlines companies in the world); he asks :
"What is the probability that, in one your flights, a passenger carryes a bomb ?".
They asked him to come back a month later and, for this second visit, his wife joined him. They meet together the chief statistician of TWA who told "according to TWA available records, there was almost one chance over 10,000".
"Thank you for the information" told the man. But he immediately added : "and what is the probability that, in one your flights, two passengers, indepently of eachother, carryes a bomb ?".
And the chief statistician said : "I am sorry; we do not have such information possibly available ! But, since you said that they are independent from eachother, then the square of the previous probability, that is to say one chance over 100,000,000, is the correct answer".
Very politely, the man thanked again and, switching to his wife, told her : " Hey, honey, do you see how clever I am carrying my own bomb everytime I travel !"
I hope that this joke makes you understanding the answers to your question !!
