Probability of Scoring 12 on a True False Exam with Randomly Selected Answers

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In summary: This is calculated by finding the total number of possible combinations for the student to choose 7 false answers out of 15 questions and 8 true answers out of the remaining 8 questions, divided by the total number of possible combinations for the student to choose any 7 questions out of 15. In summary, the problem involves calculating the probability of a student randomly selecting 7 false answers and then answering the remaining 8 questions correctly, resulting in a score of 12 out of 15.
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mitjak
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I've been struggling with this problem for two days. It is out of the Concepts in Probability and Stochastic Modelling textbook (1.4-13), and it goes like this:
A true false exam has 15 questions of which 5 are true and 10 are false. A student randomly selects seven questions and answers those false. The remainder of the questions the student answers true. The best possible score the student could obtain on this exam is 12 correct answers. What is the probability of this happening?

The problem appears in one of the introductory chapters, in this case the one on combinations and permutations, but to me it seems like the problem is a bit more complex than that. For instance, when choosing the first 7 questions, the student is guessing, so the odds of him picking the right answer are:
[tex] ({\frac{2}{3}})^7 [/tex].

The sample space doesn't change as he goes about guessing the answers, so neither combinations nor permutations apply here, it seems. But then, when he fills out the remaining 8 true answers, he's no longer randomly picking answers but just marking those remaining 8 answers as false.

I am probably overthinking the problem, but at this point I'm stuck. Would greatly appreciate any help.

EDIT: crap. Didn't realize I'm not supposed to post in this forum. Apologies. Reposting in the coursework forum.
 
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The probability of the student getting 12 correct answers on the exam is (5C7)*(10C8)/(15C7) = 0.0892.
 

Related to Probability of Scoring 12 on a True False Exam with Randomly Selected Answers

1. How accurate are guesses on a test?

The accuracy of guesses on a test can vary greatly depending on the individual and the type of test. In some cases, a guess may be correct purely by chance, while in others it may be based on prior knowledge or educated guessing. It is important to note that guessing is not a reliable test-taking strategy and should not be relied upon as a primary means of answering questions.

2. Is it better to guess or leave a question blank?

It is generally better to make an educated guess rather than leaving a question blank. Leaving a question blank guarantees a score of zero, while a guess has at least a chance of being correct. However, if there is a penalty for incorrect answers, it may be better to leave the question blank if you are unsure of the answer.

3. How can I improve my guessing skills on a test?

Improving guessing skills on a test is not recommended, as it should not be relied upon as a test-taking strategy. Instead, focus on studying and understanding the material to improve your chances of answering correctly. If you do choose to guess, try to eliminate obviously incorrect options and make an educated guess based on your knowledge and the context of the question.

4. Can I use a process of elimination to guess on a test?

Using a process of elimination can be a helpful strategy when guessing on a test. By eliminating obviously incorrect options, you increase your chances of guessing correctly. However, it is important to note that this should not be your primary means of answering questions and should not be relied upon as a sole test-taking strategy.

5. Are teachers able to tell if I am guessing on a test?

Teachers may be able to tell if you are guessing on a test if your answers are consistently incorrect or if your guesses are not logical. However, it is important to always do your best and answer to the best of your ability, rather than relying on guessing as a means of answering questions. Teachers are more likely to be impressed by well-thought-out answers, even if they are not completely correct.

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