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I need help with a probability problem and how to simulate it in a program.

A large number of people are individually asked a series of difficult questions. Statistics show that 40% of people who answered a question correctly also answered the next question correctly. 35% of those who answered a question correctly also answered the question that followed the next question correctly.

What is the probability of someone answering a question correctly, if:

    a) they answered both previous questions correctly  
    b) they answered the previous question correctly, but the one before incorrectly.
    c) they answered the previous question incorrectly and the one before correctly.
whuber
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tams
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    Is this homework? There is a certain number of questions being asked? Have you tried posing your problem using conditional probabilities? – Néstor Jun 24 '12 at 16:22
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    When you change "answer a question correctly" to "is female," "answered the previous question correctly" to "has blood type AX3," and "answered the one before the previous question correctly" to "has long hair," and alter 40% to 80% and 35% to 90%, then you are asking close variants of http://stats.stackexchange.com/questions/30842/what-is-the-probability-that-this-person-is-female. The correct answers in that thread, and some of the comments, will show you what can be known here and will indicate what additional assumptions are needed to obtain unique answers. – whuber Jun 24 '12 at 16:26
  • @whuber. I've looked a that question, and I don't think it's particularly helpful in relation to this question. In fact I had seen a similar question elsewhere. The difference with this question is that both probabilities are below 50% which, if we were to use the method from the other question, would result in a probability below 35%. With this question, there is an explicit dependability. It is implied that being able to answer a question correctly increases the probability that subsequent questions are answered correctly increases. – tams Jun 24 '12 at 16:55
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    Well, upon making the translation I suggested, I find that thread to be *exactly* the same setup (i.e., a problem about conditional probabilities of three events) and can be solved using exactly the same techniques. I suspect many others might agree with this assessment. Perhaps you could elicit some answers that are particularly helpful to you by explaining what you think the essential differences are between that question and yours. – whuber Jun 24 '12 at 16:59
  • As above. the formula ab/(ab + (1 - a)(1 - b)) would result in a probability below 35% for someone giving correct answers to the previous two questions. That just doesn't seem logical. – tams Jun 24 '12 at 17:03
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    If you rewrite your question as: "Statistics show that 60% of people who answered a question correctly answered the next question *incorrectly*. 65% of those who answered a question correctly also answered the question that followed the next question *incorrectly*." An answer less than 35% to question (a) seems reasonable, no? – Marc Shivers Jun 25 '12 at 03:26
  • I'm not sure. That's why I'm asking the question. What if we were to learn that overall, only 30% of the these difficult questions were answered correctly? Applying the non dependency formula would give us 26.4%. Also, if we are told that 40% who got question 2 correct also answered question 3 correctly, then I fail to see why the probability should drop from 40% to 26.4% because they also answered question 1 correctly. Answering questions correctly, should, in my view, enhance the probability that the third question will be correctly answered; not lower it. – tams Jun 25 '12 at 14:39
  • Tams, your intuition may be correct, but you are relying on assumptions that are not part of the question. The point of the other thread is that this question requires additional (quantitative) assumptions in order to have a unique answer. – whuber Jun 25 '12 at 14:44
  • @whuber, what are those additional assumptions? We are told that there is a certain level of dependency, unlike with hair length and blood type which had no dependency. I don't think it was implied that people grow their hair differently depending on their blood group. – tams Jun 25 '12 at 15:11
  • Do you now have a different view from what you had yesterday, when you said this was exactly the same as the other question? Are you now saying that the question has insufficient information, when yesterday you seemed pretty sure that all the information that was required was available? – tams Jun 25 '12 at 15:19
  • I'm sorry about the misinterpretation, tams: I have not changed anything. Your question had no answer yesterday and it has no answer today, either, for reasons explained separately by @Michael Chernick and myself in our answers to the related question. The additional assumptions you need concern the independence or lack thereof of the three events. – whuber Jun 25 '12 at 16:39
  • Yes, I think that it's this statement that I've somehow misinterpreted... "I find that thread to be exactly the same setup (i.e., a problem about conditional probabilities of three events) and can be solved using exactly the same techniques. I suspect many others might agree with this assessment." What you meant, apparently, is that it is not exactly the same setup, and cannot be solved without further information that you are either unwilling or unable to describe. – tams Jun 25 '12 at 17:11

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