Bayes Rule Through Examples

Learn bayes updating, base rates and false flags.

Lesson 3 of 32 parts Practice Flashcards

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Bayes Updating

Objective: Explain bayes updating, apply the stated method, and check its conditions.

Bayes’ rule gives P(A|B)=P(B|A)P(A)/P(B), with P(B)>0. Count-based examples update belief by restricting to observations consistent with B. Base rates matter because the subgroup contains contributions from different original groups, not only the desired group. Include all pathways contributing to the conditioning event.

Formulas

$P(A|B)=P(B|A)P(A)/P(B)$

Worked examples

Try itA synthetic box has 168 blue and 196 red tokens. Half the blue tokens and one quarter of the red tokens are square. One uniform draw is known to be square. Find P(blue | square).Show answer

Step 1: The square subgroup contains 84 blue squares and 49 red squares. Step 2: Conditioning divides blue-square count by all squares: 84/(84+49). Step 3: The posterior probability is 12/19; prior color frequencies and color-specific square rates both contribute.

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Bayes Updating

Objective: Explain bayes updating, apply the stated method, and check its conditions.

Formulas

$P(A|B)=P(B|A)P(A)/P(B)$

Worked examples

Try itA synthetic box has 168 blue and 196 red tokens. Half the blue tokens and one quarter of the red tokens are square. One uniform draw is known to be square. Find P(blue | square).Show answer

Base Rates and False Flags

Objective: Explain base rates and false flags, apply the stated method, and check its conditions.

An observed flag can come from a genuine target case or a false flag. The fraction of flagged cases that are genuine uses all flags as its denominator. Even a sensitive detection rule can yield many false flags when nontarget cases are common. Sensitivity alone does not equal the probability a flag is genuine.

Formulas

P(target|flag)=true flags / all flags

Worked examples

Try itA synthetic inspection flags 42 genuinely defective items and 49 nondefective items. Among flagged items, find the fraction genuinely defective.Show answer

Step 1: The flagged group includes true and false flags. Step 2: Use true flags divided by all flags: 42/(42+49). Step 3: The fraction is 6/13; the number of false flags affects the updated probability.

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