Question

A Sudoku matrix is defined as a $9\times 9$ arrary with entries from $\{1,2,3\dots \dots 9\}$ and with the constraint that each row, each column and each of the nine $3\times 3$ boxes that tile the array contains each digit from 1 to 9 exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of square in this matrix as shown. Then the probability that the square marked by ? contains the digit 3 is

• A. $\frac{1}{21}$
• B. $\frac{2}{21}$
• C. $\frac{4}{21}$
• D. $\frac{8}{21}$

Answer 1

Answer: (B)

Three squares are shown as below

digit 3 may come only in Ist and IInd rows. In second square if ? is replaced by 3 then probability is $1/3$.
Case-1: We assume that first square contains digit 3 in first row
$\therefore$ probability is $2/7$
and corresponding to it in third square digit 3 may come in II ${}^{\text{s }}$ row
$\therefore$ probability is $3/6$
Case-2: We assume that first square contains digit 3 in second row
$\therefore$ probability is $2/7$
and corresponding to it in third square digit 3 may come in $I^{5t}$ row
$\therefore$ probability is $3/6$
Hence probability $=\frac{1}{3}\times \frac{2}{7}\times \frac{3}{6}+\frac{1}{3}\times \frac{2}{7}\times \frac{3}{6}=\frac{2}{21}$