GATE DA 2026 Set 1 — Question 64
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Probability & Statistics → Discrete Random Variables → Bernoulli
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Question
Let be a matrix such that each of its elements follows distribution independently.The probability that the row-sum of the second row and the column-sum of the third column are both equal to 3 is ________ . (Rounded off to two decimal places)
Correct answer
0.08 to 0.12
Solution
Let be the matrix with entries .
Let be the sum of the second row: .
Let be the sum of the third column: .We want to find .
Note that and share exactly one element, . Let .
We can write:
, where is the sum of 4 independent Bernoulli trials.
, where is the sum of 4 independent Bernoulli trials. and are independent and both follow .
We condition on :
Case 1:
. .
. .
Contribution: .Case 2:
. .
. .
Contribution: .Total Probability = .
.Rounding to two decimal places, we get .
Let be the sum of the second row: .
Let be the sum of the third column: .We want to find .
Note that and share exactly one element, . Let .
We can write:
, where is the sum of 4 independent Bernoulli trials.
, where is the sum of 4 independent Bernoulli trials. and are independent and both follow .
We condition on :
Case 1:
. .
. .
Contribution: .Case 2:
. .
. .
Contribution: .Total Probability = .
.Rounding to two decimal places, we get .
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