How much do you expect?

Probability Level 1

Suppose you are sitting in an exam hall for an IQ-test containing 5 True/False questions and 5 multiple-choice questions , each having 4 different choices such that only one of the choices is right.

The probability of scoring 100% in the test by selecting options randomly is $\frac{p}{q}$ where $p$ and $q$ are integers.

$\left\lfloor \log _{ 2 }{ (p+q) } \right\rfloor \quad =\quad ?$

The answer is 15.

2 solutions

Zeeshan Ali
Jan 23, 2016

For a True/False question with 2 options, you can choose 1. Hence the probability of it to be right is $\frac{1}{2}$ .
For an MCQ with 4 options, you can choose 1. Hence the probability of it to be right is $\frac{1}{4}$ .

Therefore the probability to score 100% is:

$P=\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{4}\times\frac{1}{4}\times\frac{1}{4}\times\frac{1}{4}\times\frac{1}{4}=\frac{1}{2^5}\times\frac{1}{4^5}=\frac{1}{2^5}\times\frac{1}{2^{10}}=\frac{1}{2^{15}}$

Now as for the requirement we say $p=1$ and $q=2^{15}$ $\implies p+q=1+2^{15}$ , therefore: $\left\lfloor \log _{ 2 }{ (1+2^{15}) } \right\rfloor \quad =\quad 15$

Sorry, your answer is wrong. The probability of getting all answers right is dependent on the intelligence of the test-taker !!

Tony Miller - 4 years, 9 months ago

Sorry, your commentary is wrong, this the possibility no the probability

Elijah Frank - 5 months, 2 weeks ago

Exactly Same Way.

Kushagra Sahni - 5 years, 4 months ago

Actually, log 2(2^15) = 15. log 2(1+2^15) > 15 Maybe it would be good to add a line about random selection of the answers to the statement.

Sergey Grey - 4 years, 9 months ago

Correct, problem writer needs to be more accurate

Tony Miller - 1 year, 9 months ago

How do you know that the test-taker does not know any of the answers? Please specify in the problem.

Siva Budaraju - 4 years, 5 months ago

Another poorly worded question with an incorrect answer. Firstly it isn't specified that the test taker is guessing randomly. Secondly the answer is slightly bigger than 15. Who the fuck is coming up with these questions?

Aditya Dua - 3 years, 11 months ago

correct on bothcounts

Tony Miller - 1 year, 9 months ago

Thought I would add - why/how is this in the expected value quiz?

Arthur Conmy - 3 years, 11 months ago

That's incorrect, the identity is: log(a*b) = log(a) + log(b). Thus, the answer should be 15.000044.

Y Adam - 3 years ago

take floor of 15.000044 and you get 15

Zeeshan Ali - 3 years ago

I didn't read anything in you problem statement about "take floor" Zeeshan. If you want us to "take floor" , you need to say so in your problem statement. !

Tony Miller - 1 year, 9 months ago

Correct, glad you are paying attention. Problem writer should be more careful/accurate!

Tony Miller - 1 year, 9 months ago

5 question with truth/false option = 2^5, 5 questions with 4 options to answer = 2^10, 2^5+2^10 = 2^15 with log2(2^15) = 15.

Elijah Frank - 5 months, 2 weeks ago

Laurent Shorts
Apr 21, 2016

If you pick your answers randomly, of course…

I'd suppose that if you're willing to pass an IQ-test, you'd try your best ^^

How much do you expect?

The answer is 15.

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