Bayes' Rule and Updating a Belief · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Flipping what the test tells you

Computing · Machine Learning · ages 18-19
Name ______________________   Date ____________

  1. In a medical test, what does a false positive mean?

    • The test flags you but you lack the condition
    • The test clears you but you carry the condition
    • The test is run twice with one match

  2. Which fraction is the posterior P(sick given positive)?

    • All positives divided by the true hits
    • True hits divided by all positives
    • The prior plus the false alarm rate

  3. A 99 percent accurate test is right nearly every time it reports positive, even for a very rare condition.

    Circle one:   True   False

  4. In a medical test, what does a false positive mean?

    • The test says you have the condition but you do not
    • The test says you are healthy but you do have the condition
    • The test is run twice with the same result
    • The condition is so rare the test cannot work at all

  5. If a disease is very rare, a test that is 99% accurate will be right nearly every time it reports a positive result.

    Circle one:   True   False

  6. A disease affects 1 in 100 people. A test catches the disease 99% of the time when it is there, and it wrongly flags healthy people only 1% of the time. If you test positive, what is roughly the chance you actually have the disease?

    • About 50%
    • About 99%
    • About 1%
    • About 90%

  7. A prior is 0.02, the likelihood is 0.90, and the false alarm rate is 0.05. What is P(condition given positive)? Round to two decimals.

    Answer: ______________

  8. Out of 1000 people, 10 have a condition and 990 do not. A test flags 9 of the 10 sick people and also flags 49 of the healthy ones by mistake. Out of everyone flagged, what fraction actually has the condition? Round to two decimals.

    Answer: ______________

  9. After one positive your chance is 0.27. A second, separate positive with the same rates arrives. What do you do?

    • Reuse 0.27 as the new prior and run Bayes again
    • Average 0.27 with the test accuracy
    • Keep 0.27, since one update is enough

  10. A condition affects 2% of a population. A test detects it 90% of the time when present, and gives a false positive 5% of the time when absent. A person tests positive. Round P(condition | positive) to two decimals.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Flipping what the test tells you W1-mt_A4p5--f7Io-s1

  1. The test flags you but you lack the condition · Positive names the result, false names its truth: a wrong alarm.
  2. True hits divided by all positives · The top holds the true path to a positive, and the bottom holds all paths.
  3. False · Rare means the healthy crowd dwarfs the sick few, so false alarms dominate.
  4. The test says you have the condition but you do not · A false positive fires when nothing is wrong; a false negative misses something that is there.
  5. False · When the condition is rare, healthy people vastly outnumber sick ones, so most positive results are false alarms.
  6. About 50% · Even a 99% accurate test is wrong about half the time on a rare disease, because almost everyone it flags is healthy.
  7. 0.27 · Top 0.018, bottom 0.067, and 0.018 divided by 0.067 is about 0.27.
  8. 0.16 · Divide the true positives by everyone the test flagged.
  9. Reuse 0.27 as the new prior and run Bayes again · Each posterior becomes the next prior, and the second positive lifts the chance further.
  10. 0.27 · Multiply the prior by the likelihood for the top, then add both ways the test can fire for the bottom.
Worksheet · LightMySky