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Recipe

Verify or complete a joint probability table

Use when: you're given a table of joint probabilities, possibly with one entry unknown, and asked to confirm it's valid or find the missing value

Goal: either a confirmed valid joint pmf, or the missing cell's value that makes the table valid

Taught in Joint probability mass functions

Quick recipe

  1. Check every cell is nonnegative
  2. Add every cell in the table together
  3. If a cell was unknown, solve for it and check it back
  1. 1

    Look at each entry in the table; every one has to be zero or positive, with no exceptions, since a probability can never be negative.

    Example

    For a table with cells 0.10, 0.30, 0.20, 0.40: all four numbers are at least 0, so this condition holds.

    Why: a negative entry can't be a probability of anything, so it disqualifies the table immediately, no matter what the cells add to.

  2. 2

    Sum all of the entries, across the whole table, not just one row or column.

    Example

    0.10+0.30+0.20+0.40=1.00

    • every cell is a specific number — compare this total directly to 1: equal means the table is a valid joint pmf, anything else means it isn't.
    • exactly one cell is an unknown, such as k — set this total equal to 1 and solve the resulting equation for the unknown instead of comparing.

    Why: the joint pmf's second defining condition is that the probability spread across every outcome pair adds to exactly 1.

  3. 3

    When the previous step ended in an equation rather than a finished number, isolate the unknown by subtracting every known cell from 1. Then substitute the result back into the table, and confirm the whole table both sums to 1 and stays nonnegative.

    Example

    For cells 0.05, 0.15, m, 0.55: m=1-0.05-0.15-0.55=0.25. Substituting back, m=0.25 \ge 0, so the completed table 0.05, 0.15, 0.25, 0.55 is valid.

    Why: solving the equation algebraically can still hand back a negative or otherwise invalid number, so the completed table has to be checked against both original conditions, not just produced by algebra.

Common mistakes

  • Checking only the nonnegativity condition, or only the sum-to-1 condition, and stopping, when a valid joint pmf needs both to hold.
  • Summing only part of the table, such as skipping a row or a cell, when adding up the total.
  • Solving for the missing cell but never verifying it's actually nonnegative, since algebra alone can return a negative "probability."
  • Confusing the sum of the WHOLE table (this recipe's job) with the sum of a single row, which is a later section's marginal distribution, not this section's validity check.