Odd sums on every row, column and diagonal of a 3 by 3

Question

The numbers $1$ to $9$ are placed at random in the cells of a $3\times3$ grid, each used once. What is the probability that every row, every column and both main diagonals have an odd sum?

Short answer

$\tfrac1{126}$ . Only the set of cells holding the 5 odd numbers matters, and a quick check shows exactly one such set works: odds in the centre and the four edge-midpoints, evens in the corners. That is one of $\binom95=126$ equally likely sets.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Counting and Combinatorics
Conditional Probability, Total Probability, and Bayes' Rule

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