exam answer sheets

Mar 2010
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Shaniqua is taking a multiple choice exam and is going to guess all of her answers. There are 10 questions on the exam, each question with 3 possible choices and only one correct choice. What is the probability that she will make at least 60% on the exam just by guessing?

Here is as far as I have gotten:

there are 3^10 = 59049 possible ways to fill out the exam answer sheet. I know that I need to add the probabilities of getting 6/10 correct + prob7/10 + prob8/10 + prob9/10 + prob10/10

i know that the probability of getting 10/10 by guessing is 1/59049, but I dont understand how to compute the probability of the others, can someone please point me in the right direction
 

Plato

MHF Helper
Aug 2006
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Shaniqua is taking a multiple choice exam and is going to guess all of her answers. There are 10 questions on the exam, each question with 3 possible choices and only one correct choice. What is the probability that she will make at least 60% on the exam just by guessing?
\(\displaystyle \sum\limits_{k = 6}^{10} {\binom{10}{k}\left( {\frac{1}
{3}} \right)^k \left( {\frac{2}
{3}} \right)^{10 - k} } \)
 
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Mar 2010
110
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I do not fully understand the notation...we are just beginning probability and are not allowed to use sums and limits in our answers.

so from what i can tell for k=9, that would be getting 90% on the exam by guessing, would be computed as follows:

10C9 * (1/3)^9 * (2/3)^1 =

how would i compute the number of exam possibilities with 90% correct?
 

matheagle

MHF Hall of Honor
Feb 2009
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this is just a binomial with n=10 and p=1/3

You want \(\displaystyle P(X\ge 6)= P(X=6)+ P(X=7)+ P(X=8)+ P(X=9)+ P(X=10) \)

which is exactly the same as Plato's sum.
 
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