Steve guesses randomly on a 15-question multiple choice test in which each problem has two choices, a correct one and an incorrect one. What is the probability that he gets at least half of the questions correct? Express your answer as a common fraction

tertre Sep 10, 2016

#1**0 **

well, the probability of him getting the first question right is 1/2, and the probability of him getting the first and second questions correct is 1/2*1/2 and so on. since there are 15 different questions and 15 is an odd number, at least halve of 15 is [15/2]=8. so that means we have to multiply 1/2 by 1/2 eight times but doing this could be a bit tedious, so we do (1/2)^{8}=1/(2^{8})=1/256

so the probability of him getting at least halve the questions correct is 1/256

Scruffy23. Sep 10, 2016

#5**0 **

"At least half" means that he gets 8 or more correct.

Probability of getting 8 correct

C(15,8)(1/2)^8(1/2)^7 = 6435/32768

Probability of getting 9 correct

C(15,9)(1/2)^9(1/2)^6 = 5005/32768

Probability of getting 10 correct

C(15,10)(1/2)^10(1/2)^5 = 3003/32768

Probability of getting 11 correct

C(15,11)(1/2)^11(1/2)^4 = 1365/32768

Probability of getting 12 correct

C(15,12)(1/2)^12(1/2)^3 = 455/32768

Probability of getting 13 correct

C(15,13)(1/2)^13(1/2)^2 = 105/32768

Probability of getting 14 correct

C(15,14)(1/2)^14(1/2)^1 = 15/32768

Probability of getting 15 correct =

C(15,15)(1/2)^15(1/2)^0 = 1/32768

So....the total probability is = [6425 + 5005 + 3003 + 1365 + 455 + 105 + 15 + 1 ] / 32768 =

16374/32768 = 8187/16384

CPhill Sep 10, 2016

#6**0 **

that doesn't really makes sense because 8187/16384 is slightly less than one half (ok then its 0.49969482421875) so the probability of guessing at least half of the questions correct is around 0.5? it may seem ok at first but... but... but... but...

FIZZLE CRACK BANG BOOM sorry actually your right in my answer i was using the same anology as a coin toss heads in a row but this is not the same thing steve doesn't have to get them correct in a row to win he just has to them correct. sorry, sometimes my brain dissolves into molten goop it just took some time for it to solidify again.

Scruffy23.
Sep 10, 2016