Prob puzzle

The_Daddy_H

He is a pertinent puzzle. Suppose the pass rate at a driving test centre is 50%, assuming people don't get any better at driving, what is the likelihood of passing your test first time?

What is your expected number of attempts?

Now assume you get better each attempt by a factor x, what now?

Pherekydes

I know you're looking at this from a probabilistic viewpoint, but average values don't apply to individual cases. If you're a competent driver (on the day) you'll pass, if you aren't you won't.

Fremen

Off the top of my head, the expected number of attempts is two, which you can see by summing the series 1/2 + 2/4 + 3/8 + 4/16 + ...

It'd take a while to work out the second part, and I have an exam tomorrow, so I'm not too keen to do it.

But yeah, I agree with slowcoach, this problem might be better posed in terms of tossing coins or something

Grudaire

Do we include new people evey time, to replace those who pass?

MathsManiac

The question seems to me not to be well defined.

As mentioned earlier, the pass rate is not an indication of a randomly selected individual's chance of passing.

If you want people to assume that it is, you'd have to say something like this: the tester randomly decides whether to pass or fail the person, irrespective of their competence.

This of course, would undermine the intention of the last part of your question, since "getting better by a factor of x", (whatever that might mean) would have no impact on your probability of passing.

If you really wanted to model this situation properly, your best bet would be to look up a topic called "item response theory", which involves modelling the probability of success at a particular task or sequence of tasks as a function of an individual's competence (considered as a position on an ability scale).

You'd need more information about the "item response curve" for the test (as held in the test centre in question), along with some more specification about what you mean by "getting better by a factor of x".