Skip to content
Free public beta until December 4, 2026: every course is open, no card needed. After that, the first chapter of each course stays free and full access is a one-time $39 ($19 for up to 100 founding members chosen for helpful feedback). Your progress is kept. How the beta works

Recipe

Find a conditional distribution from a joint table

Use when: you're told one variable's value is given, and asked for the OTHER variable's whole distribution under that condition

Goal: a complete, valid probability distribution for the other variable, conditioned on the given value

Taught in Conditional distributions

Quick recipe

  1. Identify which row or column the given condition selects
  2. Find that row or column's own marginal total
  3. Divide each cell in the row (or column) by that total
  4. Check the resulting numbers sum to 1
  1. 1

    The event you're conditioning on picks out exactly one row or column. It's a row if it names a value of the row variable. It's a column if it names a value of the column variable. Find that row or column now.

    Example

    Conditioning on "rain" selects the rain row of the weather/commute table: [0.12, 0.08].

    • the given condition names a value of the row variable — select that row, and every division in Step 3 divides across that row's cells.
    • the given condition names a value of the column variable — select that column, and every division in Step 3 divides down that column's cells.

    Why: every probability in the conditional distribution comes only from cells consistent with the given condition, and those are exactly the cells in that one row or column.

  2. 2

    Add up every cell in the selected row (or column), the same row-summing procedure as the marginal-distributions recipe, to get the probability of the condition itself.

    Example

    P(\text{rain}) = 0.12+0.08=0.20

    ⚡ Shortcut

    If this marginal was already computed earlier in the problem, such as in a prior marginal-distributions step, reuse that number directly instead of re-summing the row from scratch.

    Why: this total is the denominator every cell in the row gets divided by, since it's the probability of the very thing being conditioned on.

  3. 3

    Take each individual cell from the selected row or column and divide it by the marginal total from the previous step, one cell at a time.

    Example

    P(\text{late}\mid\text{rain}) = \dfrac{0.12}{0.20} = 0.6, \qquad P(\text{on time}\mid\text{rain}) = \dfrac{0.08}{0.20}=0.4

    Why: dividing by the row's own total rescales it back up so it sums to 1, which is exactly what conditioning means: restricting to that row, then renormalizing.

  4. 4

    Add up every conditional probability just computed; the total must come out to exactly 1, since conditioning always produces a valid distribution of its own.

    Example

    0.6+0.4=1

    Why: dividing by anything other than the row's own exact total won't sum to 1, so this check catches a wrong denominator immediately.

Common mistakes

  • Dividing by the whole table's total, which is always 1, instead of by the selected row's or column's own total; this silently returns the original joint probabilities unchanged.
  • Selecting the wrong row or column, such as dividing by a column total when the given condition actually names a row value.
  • Stopping after computing one conditional probability instead of finding the whole distribution across every value of the other variable.
  • Skipping the final sums-to-1 check, missing a division that used the wrong denominator.