Conditional Probability - chance for an event to happen

Question

I am learning probabilities at the moment and I have come across this problem:

A person takes four tests in succession. The probability of his passing the first test is p, that of his passing each succeeding test is p or p/2 depending on whether he passes or fails the preceding one. He qualifies provided he passes at least three tests. What is his chance of qualifying?

How should I start to think about this? Intuitively I think of it as a graph and put values on edges but this does not seem very formal. I would appreciate some help. Thank you

Answer 1

Hint:

Let's say that $E_{i}$ denotes the event that he passes the $i$-th test. Then his chance of qualifying is:

$$P\left(E_{1}\cap E_{2}\cap E_{3}\right)+P\left(E_{1}^{c}\cap E_{2}\cap E_{3}\cap E_{4}\right)+P\left(E_{1}\cap E_{2}^{c}\cap E_{3}\cap E_{4}\right)+P\left(E_{1}\cap E_{2}\cap E_{3}^{c}\cap E_{4}\right)$$

Answer 2

The tree approach is perfectly fine. Writing it with fancy symbols doesn't make it more rigorous, as it just restates what is in the tree. In either case, you will need to enumerate the possibilities and get each probability.

There are exactly four ways to pass the test:

1. Get the first three tests correct
2. Fail one of the first three test and pass the last test.

(note: drhab wrote the above in theoretical terms)