Conditional Probability from a Two-Way Table: Choose the Right Total

For “A given B,” restrict attention to the B group first. Divide the number in both A and B by the number in B, provided that group is nonempty.

The condition changes the group you are counting

“Given” is not decoration. It tells you which outcomes remain possible after new information is supplied. Write P(A|B) = number in A ∩ B / number in B when selecting uniformly from a finite table. The denominator is not automatically the grand total. Reversing A and B generally changes that denominator.

Before calculating, label the table clearly and confirm that each person belongs to exactly one row and one column. Row and column totals should agree on the grand total. These bookkeeping checks prevent arithmetic errors from masquerading as probability errors.

Worked table

Forty students each choose one activity, coding or music. We also record whether they travel by bus. All students are equally likely to be selected.

Travel	Coding	Music	Total
Bus	12	8	20
Not bus	6	14	20
Total	18	22	40

Given that the selected student travels by bus, the probability of coding is 12/20 = 3/5. Given that the student chose coding, the probability of travelling by bus is 12/18 = 2/3. Both fractions use the same overlap, 12, but answer different questions.

For comparison, the probability that a randomly selected student both travels by bus and chose coding is 12/40 = 3/10. That is a joint probability, with no condition restricting the original group. Write the question in words alongside each fraction until the distinction feels natural.

Can you assume independence?

The unconditional probability of coding is 18/40 = 9/20. The conditional probability among bus users is 3/5, so these events are not independent in this table. You do not need a causal explanation to make that numerical observation. A table alone does not establish that transport caused an activity choice.

Common mistake

A probability such as P(bus|coding) is not obtained by simply reading P(coding|bus) backwards. Always identify the group after the vertical bar. If its total is zero, the elementary ratio is undefined; do not enter zero as an answer. Also do not use percentages from different row bases as though their denominators were identical.

Given ke baad wala group hi naya total hai.

Try three questions, then check the working

1. In the table, find P(music|not bus).

   The not-bus row contains 20 students, including 14 music students. The probability is 14/20 = 7/10.

2. Find P(not bus|music).

   Restrict to the 22 music students. Fourteen do not use the bus, so the probability is 14/22 = 7/11.

3. Find P(coding and not bus), with no condition.

   Use the grand total: 6 students meet both descriptions out of 40. The probability is 6/40 = 3/20.

Build the next step

• Probability Class 10 Notes: Formula, Sample Space, Coins, Dice and Cards
• Probability Class 12 Notes: Conditional Probability, Bayes’ Theorem and Distributions

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Reference and authorship

The examples and practice questions here are original, with calculations checked separately. Definitions and related background: OpenStax: Contingency Tables. Prepared with AI assistance and editorial arithmetic review; no textbook affiliation is claimed.