Question 372000: If there are 60 questions on an exam, the succes rate is 1/4. You know 24 out of 60 questions, what is the probability you will suceed on the exam if the passing grade is 50%.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The probability of  successes in  trials where  is the probability of success on any given trial is given by:
\ =\ \left(n\cr k\right\)p^k\left(1\,-\,p\right)^{n\,-\,k})
Where ) is the number of combinations of things taken at a time and is calculated by !})
Given that you know 24 out of the 60, you need the probability of guessing at least 6 out of 36 with a per question probability of  .
You can go about this one of two ways. The probability of at least 6 correct out of 36 is the probability of exactly 6 plus exactly 7 plus exactly 8 ... plus exactly 35, plus exactly 36. Symbolically:
\ =\ \sum_{i=6}^{36}\left(36\cr\ i\right\)\left(\frac{1}{4}\right)^i\left(\frac{3}{4}\right)^{36\,-\,i})
Which, trust me, is a very long, tedious, and therefore highly error prone calculation in that the sum consists of 31 terms, several of which have coefficients in the billions.
However, there is an easier way. Note that the probability of getting 5 or less correct PLUS the probability of getting 6 or more correct is just 1, or certainty -- if you guess at 36 questions it is an absolute certainty that you will get between 0 and 36 of them correct. Hence the desired probability is 1 minus the probability of getting 5 or less correct:
\ =\ 1\ -\ P_{36}\left(\leq5,\frac{1}{4}\right)\ =\ 1\ -\ \sum_{i=0}^{5}\left(36\cr\ i\right\)\left(\frac{1}{4}\right)^i\left(\frac{3}{4}\right)^{36\,-\,i})
which sum only has 7 terms (counting the 1) and the largest coefficient is 58905. Get out your calculator and get cracking.
You should get an answer that is between 91 and 92 percent.
John
My calculator said it, I believe it, that settles it
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