Guess what? I can get it right even if I don't know it

Algebra Level 4

I was giving an exam once. There were 60 questions and all had one correct answer out of four given options. I wasn't able to solve even a single question. So, I decided to guess.

But there was a problem. Professor said every wrong answer would add -1 mark to the final score and every correct answer would add +4 marks to the score. I looked at the exam with amazement and notice in each question professor has set 2 stupid options which were easy to eliminate. I eliminated all the 2 stupid options from all the 60 question.

Then I decided to guess all the 60 questions from the remaining 2 options left from all questions. Assuming that Professor is equally likely to set any of the options A, B, C or D as the correct answer, I did the math and found out I would get $4\times60\times\frac{1}{2} - 60\times\frac{1}{2} = 90$ marks on average if I decide to guess all the questions. I was delighted.

But I want to know more. I want to find $a$ and $b$ such that I can be 90% sure that my marks would lie between $a$ and $b$ .

a = 70 and b = 110 a = 60 and b = 120 a = 80 and b= 100 a = 40 and b = 140

1 solution

Ajinkya Shivashankar
Oct 17, 2016

Probability of getting a certain number $x$ of questions correct will be $\frac{60\choose x}{2^60}$ the number in exponent of denominator is 60

So we can calculate that there is 90% chance that if author decides to guess all questions, x would belong to $(24,36)$

For 24 questions correct author would score : 24*4-36=60

For 36 questions correct author would score : 36*4-24=120

So our answer is $(60,120)$

Actual calculation of the probability is quite fast if one begins with x=30 followed by x=29,31 then x=28,32 and adding all these probabilitiesand waiting for their value to just cross 0.9

Ajinkya Shivashankar - 4 years, 8 months ago

Guess what? I can get it right even if I don't know it

1 solution

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