Each question has a 0.5 probability of being answered correctly. [tex]X[/tex] follows a binomial distribution on [tex]n=14[/tex] questions and with success probability [tex]p=0.5[/tex]. So
[tex]P(X=9)=\dbinom{14}90.5^9(1-0.5)^{14-9}\approx0.12[/tex]
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Another way of looking at it: There are [tex]2^{14}=16,384[/tex] possible ways to fill out the answers to the test, right or wrong. The number of ways to get exactly 9 correct answers is
[tex]\dbinom{14}9=\dfrac{14!}{9!(14-9)!}=2002[/tex]
Then the probability of guessing correctly on 9 of the problems is
[tex]\dfrac{2002}{16384}=\dfrac{1001}{8192}\approx0.12[/tex]
