Let $A_{5\times5}$ 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)

Question

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $ X $ denote the number of defective items in the sample. If the variance of $ X $ is $ \sigma^2 $, then $ 96\sigma^2 $ is equal to _________.

Answer 1

Step 1: Understanding the Concept:
Since the matrix has 25 independent Bernoulli(0.5) entries, every one of the 25 cells is an independent fair coin flip. Only 9 of those cells actually affect the event we care about: the 4 remaining cells of row 2, the 4 remaining cells of column 3, and the single cell $A_{23}$ shared by both. The other 16 cells can be anything and do not change the probability.

Step 2: Key Formula or Approach:
Treat the 9 relevant cells as 9 independent fair coins, giving $2^9 = 512$ equally likely outcomes. Count how many of these 512 outcomes make the row-2 sum equal to 3 and the column-3 sum equal to 3 at the same time, then divide by 512.

Step 3: Detailed Explanation:
Split the count by the value of the shared cell $A_{23}$.
If $A_{23}=0$: the other 4 cells of row 2 must add to 3, which can happen in $\binom{4}{3}=4$ ways, and the other 4 cells of column 3 must also add to 3, again $\binom{4}{3}=4$ ways. That gives $4 \times 4 = 16$ favorable outcomes for this branch.
If $A_{23}=1$: the other 4 cells of row 2 must now add to 2 (since $2+1=3$), which happens in $\binom{4}{2}=6$ ways, and the other 4 cells of column 3 must also add to 2, again $\binom{4}{2}=6$ ways. That gives $6 \times 6 = 36$ favorable outcomes for this branch.
Total favorable outcomes $= 16 + 36 = 52$.
Probability $= \dfrac{52}{512} = 0.1015625$.

Step 4: Final Answer:
Rounding 0.1015625 to two decimal places gives 0.10, which matches the value found from splitting on the shared cell's value directly.
\[ \boxed{0.10} \]

Column 1		Column 2
(p)	First In First Out	(i)	Stacks
(q)	Lookup Operation	(ii)	Queues
(r)	Last In First Out	(iii)	Hash Tables

Let \(A_{5\times5}\) 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: 0.1

Solution and Explanation

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Let \(A_{5\times5}\) 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)