Conditional probability representations and expected frequencies

Probability • Probability

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How do expected frequencies relate to probabilities?

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Expected frequency equals probability multiplied by the total sample size, therefore converting proportions into counts.

Key concepts

What you'll likely be quizzed about

Definition of conditional probability

Conditional probability P(A|B) represents the probability of event A occurring given that event B has occurred. The calculation uses the joint probability of both events and the marginal probability of the given event: P(A|B) = P(A and B) ÷ P(B). The conditional probability is undefined if P(B) equals zero, therefore the given event must have non-zero probability.

Expected frequencies and sample size

Expected frequencies arise by multiplying probabilities by the total sample size, therefore converting proportions into counts. Representation as expected frequencies reduces rounding error and clarifies conditional calculations because conditional probability becomes a simple fraction of counts within a subgroup. Limiting factor: the interpretation requires an assumed or actual sample size; choosing too small a sample can produce non-integer or misleading counts.

Two-way tables (contingency tables)

Two-way tables display counts for two categorical variables with joint frequencies in interior cells and marginal totals at row and column edges. Conditional probability P(A|B) becomes the count in the cell where A and B both occur divided by the margin for B. Cause → effect: arranging counts by rows and columns isolates the given subgroup, therefore the conditional probability is a direct ratio of a cell to the corresponding marginal total.

Tree diagrams with expected frequencies

Tree diagrams show sequential events with branches labeled by conditional probabilities or expected frequencies. Converting probabilities into expected frequencies labels branches with counts, therefore the joint frequency at a final leaf becomes the product of branch fractions of the total sample size. Conditional probability P(A|B) becomes the expected frequency at the leaf for A and B divided by the expected frequency for B at the intermediate node.

Venn diagrams and conditional probability

Venn diagrams represent overlapping sets with region areas or counts for A only, B only and A∩B. Converting probabilities to expected frequencies populates each region with counts, therefore P(A|B) becomes the count in the overlap region divided by the total count in the circle for B. Limiting factor: Venn diagrams best suit a small number of sets and can become unclear for many categories.

Key notes

Important points to keep in mind

P(A|B) requires P(B)>0; never divide by zero.

Represent probabilities as expected frequencies by multiplying by sample size, therefore conditional ratios become counts over subgroup totals.

In two-way tables, always identify the correct marginal total that corresponds to the given condition before dividing.

In tree diagrams with expected frequencies, conditional probability becomes leaf count divided by node count at the conditioning stage.

In Venn diagrams, conditional probability uses the overlap count divided by the total count inside the conditioning circle.

Check totals: joint counts should sum to marginal totals and marginal totals should sum to the overall sample size.

Independence implies P(A|B)=P(A); failure of this equality indicates dependence.

Be cautious with rounding: use exact counts where possible or adjust consistently to keep totals correct.