Conditional Probability Reference

Two-Way Tables and Conditional Probability

A two-way table organizes two categorical variables so we can compare groups, compute conditional probabilities, and test independence.

Solved Examples

Use this table: 30 students play a sport and 20 do not; 18 sport players and 8 non-players take the bus.

Two-Way Table

BusNot BusTotal
Plays sport$18$$12$$30$
Does not play sport$8$$12$$20$
Total$26$$24$$50$

Fact Table

TermMeaningDenominator
Joint probabilityBoth events occur.Grand total.
Marginal probabilityOne row or column total.Grand total.
Conditional probabilityOne event given another.Total for the given condition.
Independent eventsKnowing one does not change the probability of the other.Compare conditional and overall probabilities.

Content Formulas

Joint
$$P(A\cap B)=\frac{\text{both}}{\text{total}}$$
Conditional
$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$
Independence
$$P(A\mid B)=P(A)$$

Classic Examples

Find a Joint Probability

Find $P(\text{sport and bus})$.

Solution Steps
  1. Locate the cell where both conditions meet.
  2. Use the grand total as denominator.
  3. Simplify the probability.
$$\text{sport and bus}=18$$$$P=\frac{18}{50}$$$$\boxed{\frac9{25}}$$

Find a Conditional Probability

Find $P(\text{bus}\mid\text{plays sport})$.

Solution Steps
  1. Restrict the sample space to students who play a sport.
  2. Count bus riders within that row.
  3. Divide by the row total.
$$\text{given sport: total }30$$$$\text{bus among sport players}=18$$$$\boxed{P(\text{bus}\mid\text{sport})=\frac{18}{30}=\frac35}$$

Find a Marginal Probability

Find $P(\text{bus})$.

Solution Steps
  1. Use the bus column total.
  2. Use the grand total as denominator.
  3. Simplify the result.
$$\text{bus total}=26$$$$P(\text{bus})=\frac{26}{50}$$$$\boxed{\frac{13}{25}}$$

Test Independence

Are playing a sport and riding the bus independent?

Solution Steps
  1. Compare the conditional bus probability for sport players.
  2. Compare it to the overall bus probability.
  3. State whether the probabilities match.
$$P(\text{bus}\mid\text{sport})=\frac35$$$$P(\text{bus})=\frac{13}{25}$$$$\boxed{\frac35\ne\frac{13}{25};\text{ the events are not independent.}}$$