Question

Let \( P = \begin{bmatrix} 1 & 2 & 3 \\4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \) be a matrix. Three elements of this matrix P are selected at random. A is the event of having the three elements whose sum is odd. B is the event of selecting the three elements which are in a row or column. Then \( P(A) + P(A|B) = \)

• A. 221/420
• B. 17/21
• C. 21/20
• D. 3/2

Hint:

For small sample spaces, listing outcomes is faster.

Answer 1

The Correct Option is B

To solve this problem, we need to understand the probability of two events: \( A \) and \( B \). 

First, let's analyze the matrix:

1	2	3
4	5	6
7	8	9

The numbers in the matrix are: 1, 2, 3, 4, 5, 6, 7, 8, and 9. We need to select three elements from this matrix.

Step 1: Calculate Probability \( P(A) \)

\( A \) is the event where the sum of the selected three elements is odd.

• Sum of three numbers is odd if 1 or 3 numbers are odd.
• There are 5 odd numbers (1, 3, 5, 7, 9) and 4 even numbers (2, 4, 6, 8).

To find \( P(A) \):

• Case 1: All three numbers are odd. Choose 3 out of 5 odd numbers: \( \binom{5}{3} = 10 \).
• Case 2: Two even numbers and one odd number. Choose 2 out of 4 even and 1 out of 5 odd numbers: \( \binom{4}{2} \times \binom{5}{1} = 6 \times 5 = 30 \).

Total favorable outcomes for \( A \): \( 10 + 30 = 40 \).

Total ways to choose any 3 numbers from 9: \( \binom{9}{3} = 84 \).

\( P(A) = \frac{40}{84} = \frac{10}{21} \).

Step 2: Calculate \( P(A|B) \)

\( B \) is the event of selecting three elements that are in a row or column.

• Rows: 3 options (each row).
• Columns: 3 options (each column).

Total ways to form rows or columns: 3 (rows) + 3 (columns) = 6.

Analyze each row/column:

• Row 1: Sum is 6; Row 2: Sum is 15; Row 3: Sum is 24.
• Column 1: Sum is 12; Column 2: Sum is 15; Column 3: Sum is 18.

Only Row 2 and Column 2 have odd sums.

Total favorable outcomes for \( (A|B) \): 2 (Row 2 and Column 2).

\( P(A|B) = \frac{2}{6} = \frac{1}{3} \).

Step 3: Calculate \( P(A) + P(A|B) \)

\( P(A) + P(A|B) = \frac{10}{21} + \frac{1}{3} = \frac{10}{21} + \frac{7}{21} = \frac{17}{21} \).

So, the answer is 17/21.