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Chapter 2 Practice Question 4

A

angelmathew

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In the question 2.4, part (i)-b, we need to find the probability that they pass the fifth exam, given that they fail the first three. So, I was expecting the only possible path would be F -> F -> F -> P -> P. But the answer and explanation are not aligning with my understanding. Could someone please explain why there are two paths given in the explanation. Is it because the question hasn't mentioned anything about the fourth exam and it can be either pass or fail?
 
Hi angelmathew

That's right, the fourth exam could be pass or fail. Also, because the next result only depends on the most recent past result, we only need to consider the final failure as the starting point. So, starting from the third failure, the paths are:

F -> F -> P; or
F -> P -> P

Hope that helps!
 
That definitely helps. Thanks a lot! One follow-up question though. In this example, the next result depends only on the most recent past result. In that case, even in part (i)-a, why don't we consider only the final pass to fail transition, that is only (1-alpha). When we write the probability as a conditional probability, only the final transition would matter due to its Markov property, right? So, why do we need to consider the entire path in that case? How can we identify in a question whether it requires us to consider the whole path or only a part of it when the process has Markov property?
 
You are absolutely right of course that the Markov property tells us that probability of failing the seventh exam only depends on the result of the sixth exam. However, this question asks for the probability that the first exam failed is the 7th. This means we have to explicitly consider the paths through the state space that meet this condition.

As a general rule of thumb, if the question asks for a probability that doesn't seem like a standard transition probability (ie it is not the probability of going from one state to another in a set number of steps), then I'd try considering different possible sample paths through the state space that meet the given requirements.

In this case, for example, it is not the 6-step transition probability from pass (passing the first exam) to fail (failing the seventh), as this would include all possible paths that mean another exam could be failed before the seventh.

Another way to think about this is by considering the following probabilities:

P(fail 7 | pass 1,2,3,4,5,6) = P(fail 7 | pass 6) = (1-a)

as you point out.

However, we also have:

P(fail 7 | pass 1, 2, 3, fail 4, 5, pass 6) = P(fail 7 | pass 6) = (1-a)

In the second scenario the 7th exam is not the first exam that they failed (as they also failed 4 and 5).

So, the (1-a) is only considering the final transition, it tells me the probability of failing 7 given I pass 6 but we want the probability that 7 is the first fail.

Hope that helps!
 
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