Ace Pre Gate 2017

Question

All elements of a 2x2 matrix "A" can have values either 0 or 1. The probability that any element gets a value (0 or 1) is 1/2. If all elements of this matrix are chosen at random, what is the probability that the determinant of this matrix is positive?

Comments

13/16 ?

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My answer was 3/16. Consider matrix as A11 = a, A12 = b, A21 = c, A22 = d

So the |A| will be positive if ad - bc > 0 that is ad > bc

which is possible when (a,d) is (1,1) and (b, c) can be (0,0) (0,1) (1,0) but not (1,1)

Hence 3 favorable outcomes out of 16 combinations.

Hence P(|A| is positive) = 3/16 = 0.1875

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Consider a=0, d=0, b=1, c=0 or b=0, c=1.

You will get many more cases

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And 0 is also positive, so your condition should be ad-bc >= 0

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0 is not positive. Neither is it negative. Hence I did not consider 0 combinations in "ad"

positive means > 0 strictly

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okay!! the question is interesting now!

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Positive should also include 0

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If they had asked "not negative" then we could have considered 0, not otherwise. See positive number definition below 

https://simple.wikipedia.org/wiki/Positive_number

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0 should not be there .!
ad-bc>0
so ad>bc
LHS of above will be 1 when both a and d are 1 similalry RHS should be zero to satisfy the condition for RHS we have three possible cases as:

11>00

11>01

11>10
total three cases out of 16 diffferent combination so 3/16 should be answer.

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Except for the "<" signs, your answer is correct

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edited now .

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i too think the answer should be 3/16

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@Kunal.

ad - bc $\neq$ 0

ie ad $\neq$ bc

Now, 2 cases:

Case 1:  ad =1 and bc=0

Thus, a =1 and d=1 and (b=0 or c=0)

Thus, we get 3 cases from here.

Case 2: ad=0 and bc=1

Thus, b=1 and c=1 and (a=0 or d=0)

Thus, we get 3 cases from here.

Thus, favourable events = 6

Sample space = 2⁴ = 16

Thus, probablity = 6/16

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@sushant question says determinant should be positive so why have you considered the second case ad < bc?

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oops....got it :)

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I think 12 cases are there when the output of the determinant is positive considering determinant may be zero.
why not
3/4