Conditional probability tables

Objective. Extract conditional probabilities from tables and diagrams.

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Learn: the key idea

Conditional probability tables (two-way tables) organise joint data by two categories so you can find probabilities that depend on a given condition. The rule is P(A | B) = (frequency of A and B) ÷ (total frequency of B), using the row or column total for the condition, not the grand total. For example, if a table shows 15 boys play football and 25 boys in total, then P(football | boy) = 15 ÷ 25 = 0.6. Always identify which row or column represents the 'given' condition first, since this restricts your denominator to that subset only, then read the matching cell as the numerator.

Worked example

A table shows 18 girls like maths out of 30 girls surveyed in total. Find P(likes maths | girl) from the table.

1 $Identify the condition: girl, with row total 30$
2 $Identify the joint frequency: likes maths and girl = 18$
3 $Apply P(A | B) = joint \div condition total = 18 \div 30$
4 $Simplify 18/30 to 3/5$

Answer: 3/5

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Common mistake. Students often divide by the grand total instead of the condition's total; e.g. using 15 ÷ 60 (all pupils) instead of 15 ÷ 25 (all boys) for P(football | boy), which understates the true conditional probability.

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