Question 296701
Assume it was only 3 answers on the test and the student needed to get at least 2 out of 3 correct.


Probability of getting exactly 0 wrong would be .5^3 = .125 * 1 = .125
Probability of getting exactly 1 wrong would be .5^3 = .125 * 3 = .375
Probability of getting exactly 2 wrong would be .5^3 = .125 * 3 = .375
Probability of getting exactly 3 wrong would be .5^3 = .125 * 1 = .125


Total probability is equal to 1 as it should be.


Probability of getting 0 or 1 wrong would be .375 + .125 = .5


Since 0 or 1 wrong is the same as getting 2 or 3 right, then this is the probability that the student will get at least 2 right.


The individual probabilities are multiplied by the number of ways they can occur.


If we let 0 = wrong and 1 = correct, then:


You can get 0 wrong only 1 way (111)
You can get 1 wrong 3 ways (110) (101) (011)
You can get 2 wrong 3 ways (001) (010) (100)
You can get 3 wrong 1 way (000)


The same concept applies to the larger numbers.


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With 10 answers, this is what happens:


p(0) = probability of getting exactly 0 correct.
p(1) = probability of getting exactly 1 correct.
etc.


p(0) = .5^10 = .000976563 * 1 = .000976563
p(1) = .5^10 = .000976563 * 10 = .009765625
p(2) = .5^10 = .000976563 * 45 = .043945313
p(3) = .5^10 = .000976563 * 120 = .1171875
p(4) = .5^10 = .000976563 * 210 = .205078125
p(5) = .5^10 = .000976563 * 252 = .24609375
p(6) = .5^10 = .000976563 * 210 = .205078125
p(7) = .5^10 = .000976563 * 120 = .1171875
p(8) = .5^10 = .000976563 * 45 = .043945313
p(9) = .5^10 = .000976563 * 10 = .009765625
p(10) = .5^10 = .000976563 * 1 = .000976563


Total probability equals 1 as it should.


Probability of getting 0 or 1 or 2 wrong is equal to:


.000976563 + .009765625 + .043945313 = .0546875


Since the probability of getting 0 or 1 or 2 wrong is the same as the probability of getting 8 or 9 or 10 right, then the probability that the student will get at least 8 correct is equal to .0546875.


The number of ways each percentage can be achieved is given by the formula:


{{{(n!) / ((x!)*(n-x!))}}}


For example, with 10 answers, the number of ways of getting exactly 4 wrong is equal to:


{{{(10!)/((4!)*(6!)) = (10*9*8*7*6!)/(4*3*2*1*6!) = (5040*6!)/(24*6!) = 5050/24 = 210}}}


With 10 answers, the number of ways of getting exactly 6 wrong is the same as getting exactly 4 wrong as shown below:


{{{(10!)/((6!)*(4!)) = (10*9*8*7*6*5*4!)/(6*5*4*3*2*1*4!) = (151200*4!)/(720*4!) = 151200/720 = 210}}}