GATE 2026 DA – Question 64
Let $A_{5 \times 5}$ be a matrix such that each of its elements follows $Bernoulli(p = 0.50)$ 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*)
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Correct answer: 0.09 to 0.11
Explanation
The second row and the third column share the element $A_{23}$. Condition on it, with probability $\frac{1}{2}$ each. If $A_{23} = 1$, the other 4 elements of the row must hold two 1s, which has probability $\frac{\binom{4}{2}}{16} = \frac{6}{16}$, and the same holds for the column. If $A_{23} = 0$, the other 4 elements must hold three 1s, with probability $\frac{\binom{4}{3}}{16} = \frac{4}{16}$ each. The probability is $\frac{1}{2}\left(\frac{6}{16}\right)^2 + \frac{1}{2}\left(\frac{4}{16}\right)^2 = 0.0703 + 0.0313 = 0.1016$, which is 0.10.