How Do You Calculate Probabilities for Guessing on a Multiple Choice Test?

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Homework Help Overview

The discussion revolves around calculating probabilities for a ten-question multiple choice exam where each question has five possible answers. The original poster seeks to find the probability of getting at least 9 questions correct while being restricted from using the binomial probability formula. Additionally, they are interested in determining the probability of getting exactly two questions correct.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of the binomial formula for calculating probabilities, with some suggesting it may be necessary for both parts of the question. Others question the rationale behind avoiding certain methods and explore alternative ways to approach the problem, particularly for calculating the probability of getting exactly two questions correct.

Discussion Status

The conversation is ongoing, with various participants offering insights into the problem. Some have provided guidance on how to approach the calculations, while others express confusion about the restrictions imposed on the methods used. There is a mix of interpretations regarding the requirements of the problem.

Contextual Notes

Participants note the challenge of calculating probabilities without using the binomial theorem, raising questions about the reasoning behind such restrictions. The discussion also touches on the distinct possibilities for scoring based on the number of questions and possible answers.

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Homework Statement




A ten question multiple choice exam is given, and each question has 5 possible answers. Pascal Gonyo takes this exam and guesses at every question. Find the probability that he will get at least 9 questions correct. Without using the binomial probability formula, determine the probability that he gets exactly two questions correct.

Homework Equations


P(at least 9) + P(less than 9) = 1



The Attempt at a Solution



P = 1-P(8)
I'm stuck on this. I found the probability of at least one question correct using this formula but I can't get it to work for 9.
 
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Use the binomial formula for the first part of the question.
 
The probability he will get exactly two questions correct is pretty easy. What is it? As is the probability he will get at least 9 correct is the sum of the probabilities of getting getting exactly 9 correct plus the probability of getting exactly 10 correct. What are they? I'm not sure I see a way to do these without using at least a vestige of the binomial theorem. But someone may prove me wrong tomorrow.
 
The binomial formula will give me exactly 9 right? how do I get at least 9?
 
Add the prob of getting exactly 9 and the prob of getting exactly 10.
 
Thanks, can you tell me how to find the probability of getting 2 correct without the binomial formula?
 
It's a dumb question! Why on Earth should they tell you not to use a particular theorem?
 
OK, so how many distinct possibilities are there for scoring?

e.g. for 3 Q's there's 8 possibilities

YYY
YYN
YNY
YNN
NYY
NYN
NNY
NNN

For 10 Q's how many distinct possibilities are there?

Oops- there's 5 possible answers per Q. Well- can you work it out for that case?
 
christianjb said:
It's a dumb question!

I am quite sure the OP is just looking for some clarity, don't be that harsh.o:)
 
  • #10
:wink:
christianjb said:
It's a dumb question! Why on Earth should they tell you not to use a particular theorem?
Perhaps because they want people to think rather than just apply formulas? It wouldn't make sense to work it out from first principles for, say, 4 or 5 out of 10, but for "exactly 2" it shouldn't be too much to expect.

What is the probability of "CCIIIIIIII" where C= correct and I= incorrect?
Can you show that the probability of 2 correct and 8 incorrect is exactly the same no matter what order?
How many different orders are there- in how many ways can you write 2 Cs and 8 Is?
 
  • #11
Good grief!

Threetheoreem: I was criticizing the Q, not the poster. The poster presumably didn't compose the question.

HOI: I said it's a dumb question because I don't like the idea of restricting how a student solves a problem. Give me the Q- but don't tell me how to find the A. It also takes intelligence to see if the binomial theorem is applicable in this case.
 

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