Common life and Maths

I'm kind of tired of hearing: "I want to be a lawyer, why would I have to study maths?" or "when I'm going to use this crap?!!, I hate maths!". Well, today I'm going to show you an example of why maths are important for common life!

I often go to work by taxi with four people, usually the treat is: each passenger pays 8 dollars, no matter where he gets out the taxi. But the other day the driver said: "I believe the treat is not fare, I think the best way to pay the trip is: when a person arrives to his place he pays what the taximeter says divided by the number of people on the cab (including me)". Everyone agreed, except me, I was kind of confused, I just kept quite and say nothing, I thought: if everyone agree it must be a good treat. But then I thought: "The price of the entire trip is usually more than 20 dollars but never more than 60 so, Is a good treat for the cab driver?"

Let's say the real cost of the trip is:

Each Lambda is the increment of the trip cost each time a person gets out the taxi. On the other hand, the taxi driver receives:

So the difference is:

If we normalize this function:

Now let's analyse the possible set of values:

If we consider that the taxi driver succeeds if he receives the real cost of the trip or more than it, we have

With this function we can count the number of times the difference is zero or is positive, so we calculate the probability of success:

So the taxi driver has low probability of wining more money....... Only if he knew something about maths!