Sample exam 2 from garp for level 1 - question 6

[sarita]

Hi David,

Hope you are well. Would you please help me understand this question better? what formula is used to find the answer?


Q - Suppose that a quiz consists of a 10 true-false questions/ A student has not studied for the exam and just randomly guesses the answers. What the probability that the student will get at least three questions correct?

a. 5.47%
b. 33.66%
c. 78.62%
d.94.53%

I don't understand the Garp answer and the formula being used. The answer is d.

many thanks in advance.
S

 

[David Harper CFA FRM]

Hi Saray,

I hope you are well, also!

As the binomial distribution is discrete, for a T/F exam with ten questions there are only 11 outcomes: 0/10 correct, 1/10 correct, .... 10/10 correct.
The P[3 or more correct] = 100% - P[2 correct] - P[1 correct] - P[0 correct]

So GARP's math is solving for P[c=0] + P[c=1] + P[c=2] under the binomial http://en.wikipedia.org/wiki/Binomial_distribution

where p[c = 2 | n = 10] = "10 choose 2" * 0.5^(2) * 0.5^(10-2) = 10C2*0.5^10;
so they took a shortcut by taking advantage of the fact that if p = 50%, then:
p^k*(1-p)^(n-k) = 50%^k*50%^(n-k) = 50%^n; in this case, 0.5^10

P[c=0] + P[c=1] + P[c=2] = 10C0*0.5^10 + 10C1*0.5^10 + 10C2*0.5^10 = (10C0 + 10C1 + 10C2)*0.5^10
e.g., 10C2 is a combination: how many ways can 2 objects be selected from 10 without regard to order

Hope that helps and i hope studies are going well, David

 

[sarita]

Many thanks David. it makes sense.. i tried to redo this after and unfortunately took me a little over 2 minutes which makes me worried about running out of time to do similar questions on the exam.. Regards, sara

 

[David Harper CFA FRM]

Sure, but that is good-to-great time on this question IMO. Not all questions will be created equally, some will be like this and some will be quick; this is a time-consuming question for pretty much everybody, unless I guess you've really, really practiced the binomial...David