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Math Help - Probability of getting at least one correct answer, with 2 out of 5 correct options

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    Probability of getting at least one correct answer, with 2 out of 5 correct options

    This is a real world problem we're trying to solve, not a test question, so in a way, your help is all the more appreciated!

    A student taking a test is presented with a question with five options. Among the 5 options presented, 2 of the 5 represent correct answers. (So, for example, if A, B, C, D and E are the options, it may be that B and E are correct.)

    They may choose up to 2 options; they can opt to select only 1 option, but may select 2. They may not select more than 2. (Essentially, think of 5 check boxes: they may not select more than 2, but they do have the option of selecting only 1.)

    We know that the probability of selecting both options correct (i.e., marking two check boxes, and both selections are correct) is represented by the following:

    \frac{2}{5}\times\frac{1}{4}=\frac{1}{10}

    Here's what we need help with. Please remember that they do have the option of only marking 1 out of the 5 check boxes.

    1. What is the probability that they will get only 1 option correct (that either (a) they mark only 1 check box, and it is a correct one, or (b) they mark 2 check boxes, and 1 is right, but 1 is wrong)?
    2. What is the probability that they will get at least 1 option correct (that either (a) they mark only 1 check box, and it is a correct one, (b) they mark 2 check boxes, and 1 of the 2 is correct, or also (c) they mark 2 check boxes, and both are correct)?


    Essentially, we need to know the different probabilities for getting only 1 correct, versus getting at least 1 correct ... and we don't know how to account for their option to only mark 1 checkbox, not two.
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    MHF Contributor alexmahone's Avatar
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    You've posted 6 problems all on the same topic of binomial probability distribution. Your title: This is a real world problem we're trying to solve, not a test question, so in a way, your help is all the more appreciated! sounds awfully suspicious.

    Maybe you should post some working if you want to get help.
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    My title may sound suspicious, but it is true. I'm not strong in probability; I know that...
    • the probability of a single event is the number of possibilities that meet my criteria over the total number of possibilities
    • the probability of two events BOTH happening is the single probability of each, multiplied together
    • the probability of one or the other of two events happening is the single probability of each, added together, minus the combined probability of both


    That is the FULL extent of my understanding of probability. That's why I have not shown much work (though, in fairness, I did show that I understood the probability of getting both correct). I've gone as far as I can.

    As to why I posted 6 similar questions: I need to be able to determine various probabilities relating to similar scenarios. The instructions here at the forum tell us not to post more than 1-2 questions per thread. Therefore, I created additional threads for the different questions I have relating to the same scenario.

    Specifically, I am creating a test which may have one of two different formats. It may be a standard, 4-option multiple choice test. Or, the questions may have 5 options, with the student able to select 1 or 2 options, but not more than 2.

    In order to determine what is the best route to take, I need to know...
    • The probability of getting ONLY one correct in the 2 out of 5 format
    • the probability of getting AT LEAST 1 correct in the 2 out of 5 format
    • the probability of each of the above in a 2 out of 5 format when the student is able to eliminate wrong answers
    • the probability of getting a passing score of 70% or more on a standard multiple choice test, when guessing
    • the probability of getting a passing score of 70% or more on a standard multiple choice test, when guessing, if they can eliminate wrong answers
    • the probability of getting a passing score of 70% or more on the 2 out of 5 format, when guessing
    • the probability of getting a passing score of 70% or more on the 2 out of 5 format, when guessing, if they can eliminate wrong answers


    Seeing the probabilities of each of these scenarios will enable me to choose how to format my test.

    Because I that comes out to quite a few different probability questions, even though they're all related to the same 2 basic scenarios, I created several separate topics for them, so as not to have too many questions in one topic... as instructed.
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    MHF Contributor alexmahone's Avatar
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by jbwtucker View Post
    My title may sound suspicious, but it is true. I'm not strong in probability; I know that...
    • the probability of a single event is the number of possibilities that meet my criteria over the total number of possibilities
    • the probability of two events BOTH happening is the single probability of each, multiplied together
    • the probability of one or the other of two events happening is the single probability of each, added together, minus the combined probability of both


    That is the FULL extent of my understanding of probability.
    You need to read a chapter on binomial probability distribution to even begin answering these problems.

    I'm not buying that this is a real life scenario.
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Right. Clearly, I'm not capable of doing this math, or I wouldn't be here. Furthermore, I'm not capable of understanding a chapter on binomial probability distribution, at this point.

    That's why I'm here. Asking for help. While I clearly don't know how to do this, it seems like perhaps you do. Can you help me? Please?
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by alexmahone View Post
    I'm not buying that this is a real life scenario.
    What else would it be?

    Look, I'm a guy who doesn't know how to do this kind of math, looking for help with this kind of math. I found a math help forum. It shouldn't be that suspicious. I have a real life scenario for which I need to know probabilities, and I don't know probability beyond the most basic concepts. So I'm here, asking for help...
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    MHF Contributor alexmahone's Avatar
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by jbwtucker View Post
    What else would it be?
    From another of your posts: The student is able to eliminate two options he knows to be incorrect, leaving him with three options, two of which are correct.

    If this was a real life scenario, how in the world would you know this?
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    No, that's not the real life scenario.

    The real life scenario is that I really do need to create this test. That's my scenario. In order to know which test to choose, I need to understand how a student should be expected to do (a) when guessing randomly, and (b) when they're able to eliminate some answers they know to be wrong.

    To answer your question, I do not know that the students will be able to eliminate 2 out of 4 or 5 options on a multiple choice test. Here's what I do know. I know the test will be administered to students who have been taught the knowledge and skills in this area; that is, it will be a test that is essentially at the level at which they have been taught. I also know that these are hard-working, diligent students who have put forth a lot of effort. So it is likely that these students, having a decent amount of knowledge of and exposure to the subject matter, will at least be able to eliminate some answers.

    Do I know that they will be able to eliminate exactly 2 options on every single question? Absolutely not. But I can't think of how you could possibly estimate, mathematically, how many options they will actually be able to answer, per question or on average. If you could, I'm sure it would be extremely advanced and difficult math. So instead, I've made an estimate. It seems likely to me that many of these students, being of the temperament and education/experience that I described above, will be able to eliminate at least one option, and more likely two. Thus, I would like to know how such a student would do.

    The bottom line is that my supervisor wants me to eliminate, to the greatest extent possible, the possibility of a student passing a test simply by guessing. We are trying to get at how much they actually know. In essence, therefore, I need to use the test that will result in the lowest score when guessing.

    That's why I need to know this information in order to, as I've said, make the decision as to how to format my test. I need to be able to say, "See, people who guess in this format have a lower chance of success than those who guess in that one; and people who eliminate answers they know to be wrong to make a so-called educated guess have a lower chance of success in this format than they do in that one."

    I don't know how many options a student will be able to eliminate per problem, and I don't need to know that. I need to be able to understand how a certain test format will perform in different scenarios. Therefore, I have come up with a few scenarios, and need to find the probabilities for those scenarios.

    The real world application is my need to create this test, to use a format that reduces a student's chance of success by guessing, and to be able to prove, via probabilities related to a few scenarios, that the format I've chosen does in fact do that.
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    If what you say is true, you are posting on the wrong forum.
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    It is true, and I'll gladly post in the right one. Which one would that be?
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by jbwtucker View Post
    It is true, and I'll gladly post in the right one. Which one would that be?
    I said that you're posting in the wrong forum (not wrong subforum). I don't know which is the 'right forum' for this kind of thing.
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    I need help with math, and I'm at a math help forum. It's even called Math Help Forum.

    Anyone else want to help me with some math?
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by jbwtucker View Post
    I need help with math, and I'm at a math help forum. It's even called Math Help Forum.

    Anyone else want to help me with some math?
    What probability have you studied? Do you have an instructor you can get advice from during or out of class? What background do you have? Where has the question come from (from a book, invented by you, set by a teacher?)

    The help you need is probbaly too broad at the moment. What have you tried?
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by mr fantastic View Post
    What probability have you studied? Do you have an instructor you can get advice from during or out of class? What background do you have? Where has the question come from (from a book, invented by you, set by a teacher?)

    The help you need is probbaly too broad at the moment. What have you tried?
    Bizarre... I thought I answered this. Hmmm.

    I have not studied probability; that is, I have not taken a class. I am not currently in a class, and I do not have an instructor. I am a middle school math teacher, focusing primarily on Algebra. I am not a math major; I simply understand middle school and high school math well enough to pass the certification test at those levels. My command of high school Algebra is very good. My understanding (recollection) of high school Trig and Calculus is moderate, as I don't teach at those levels. I don't do probability and statistics, and have never taken a class in either.

    The question came from me, in the sense that it is how I am attempting to evaluate a real-life scenario. I described that above, and I believe also in a PM to you. It's just an actual problem I'm dealing with right now.


    Moderator edit: The following pm from the OP gives additional information.

    I seem to be getting that question a lot.

    Just an actual situation I'm dealing with. The scenario I describe is not hypothetical I have to create a test, and understanding these various probabilities will help me know what format of test to create. The idea is to create a test that will minimize a student's odds of succeeding purely by guessing; and then, as a secondary priority and to the extent possible, to also minimize a student's odds of succeeding by eliminating incorrect questions and then guessing. The reason for these goals is to minimize, as much as possible, the extent to which test-taking skill and pure (good) luck are reflected in the results, such that the results will more accurately reflect what a student actually knows.

    That's my context, and I need to make a data-driven decision.
    Last edited by mr fantastic; July 28th 2011 at 08:03 PM.
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    Re: Probability of getting at least one correct answer, with 2 out of 5 correct optio

    Quote Originally Posted by jbwtucker View Post
    Bizarre... I thought I answered this. Hmmm.

    I have not studied probability; that is, I have not taken a class. I am not currently in a class, and I do not have an instructor. I am a middle school math teacher, focusing primarily on Algebra. I am not a math major; I simply understand middle school and high school math well enough to pass the certification test at those levels. My command of high school Algebra is very good. My understanding (recollection) of high school Trig and Calculus is moderate, as I don't teach at those levels. I don't do probability and statistics, and have never taken a class in either.

    The question came from me, in the sense that it is how I am attempting to evaluate a real-life scenario. I described that above, and I believe also in a PM to you. It's just an actual problem I'm dealing with right now.


    Moderator edit: The following pm from the OP gives additional information.

    I seem to be getting that question a lot.

    Just an actual situation I'm dealing with. The scenario I describe is not hypothetical I have to create a test, and understanding these various probabilities will help me know what format of test to create. The idea is to create a test that will minimize a student's odds of succeeding purely by guessing; and then, as a secondary priority and to the extent possible, to also minimize a student's odds of succeeding by eliminating incorrect questions and then guessing. The reason for these goals is to minimize, as much as possible, the extent to which test-taking skill and pure (good) luck are reflected in the results, such that the results will more accurately reflect what a student actually knows.

    That's my context, and I need to make a data-driven decision.
    In my experience, a multiple choice test with a sufficient number of well written questions with appropriately chosen 'wrong options' and disatrctors will more than satisfy your criteria. A well constructed and written test is what you need to focus on, not a probability analysis.
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