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Bayes’ Theorem Flashcards: Base Rates & Evidence Updates

Practice 48 original Bayes theorem flashcards: build source-and-evidence weights, normalize across all sources, repair base-rate mistakes, and check sequential updates.

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A signal can be common for one source without making that source likely after the signal. These 48 original Bayes theorem flashcards help you keep the source share, source-specific likelihood, total evidence, and posterior separate.

Start with priors and likelihoods. Then calculate prior × likelihood branch weights for fictional bags, batches, routes, and sensors; add every possible source branch to get the evidence total; and normalize the requested source weight. Two-source updates lead into three-source comparisons and subset posteriors. Short repair prompts target reversed conditions, missing priors or branches, the wrong complement, and branch weights mistaken for posteriors. Six cards handle sequential evidence, including when conditional independence permits multiplication and when a repeated display supplies no new observation.

The sequence introduces each prerequisite before its dependent calculation, then interleaves calculations, interpretations, and repairs. Related variants are spaced so one answer does not directly cue the next; the app’s scheduler handles later review spacing. Each card gives the model assumptions it needs and an explained answer.

This is a focused introductory probability set, not a full statistics or exam course. It avoids two-way-table count drills already covered by the separate conditional-probability deck, mechanical reverse calculations that cannot recover unknown inputs, continuous Bayesian models, odds-form algebra, medical and financial decisions, and claims about real-world sources from toy assumptions. All examples, wording, order, and cover are original. Mathematical facts were checked against NIST’s Engineering Statistics Handbook and MIT OCW’s probability lecture; their prose, exercises, and figures were not copied. AI assisted drafting, followed by individual factual and numerical review. The CC0 label covers only the original expression and generated artwork to the extent rights exist.

Cartes de ce paquet

  1. Carte 1

    Question

    For a Bayes update over hidden sources, what must the listed sources cover?

    Réponse

    Every possible source, exactly once. The sources must be exhaustive and mutually exclusive so their evidence branches can be added.

  2. Carte 2

    Question

    Before seeing a result, a token comes from bag A with probability 0.3. What is 0.3 called in a Bayes update?

    Réponse

    The prior for bag A. It is A’s source probability before the result is observed.

  3. Carte 3

    Question

    A token from bag A is blue with probability 0.8. In a Bayes update after seeing blue, what role does 0.8 play?

    Réponse

    The likelihood of blue given A. It is P(blue | A), not the chance that a blue token came from A.

  4. Carte 4

    Question

    An observed signal can come from every possible source. What probability belongs in the denominator of a source posterior?

    Réponse

    The total probability of the signal. Add its prior × likelihood branch weight across all possible sources.

  5. Carte 5

    Question

    Only bags A and B can be chosen. P(A) = 0.3 and P(red | A) = 0.8. What is the joint weight for A and red?

    Réponse

    0.24. Multiply 0.3 × 0.8; this is P(A and red), before normalization.

  6. Carte 6

    Question

    A bag yields blue with probability 0.8. A learner calls 0.8 the chance the bag was chosen after seeing blue. What is the mistake?

    Réponse

    The condition was reversed. 0.8 is P(blue | bag); P(bag | blue) also needs source priors and the other bags’ blue rates.

  7. Carte 7

    Question

    Only batches A and B supply parts. Their shares are 0.25 and 0.75; a part is marked with probabilities 0.6 and 0.2, respectively. What is P(marked)?

    Réponse

    0.30. Add both marked branches: 0.25 × 0.6 + 0.75 × 0.2 = 0.15 + 0.15.

  8. Carte 8

    Question

    A source makes 70% of its tokens striped and supplies 20% of all tokens. A learner uses 0.70 as its striped branch weight. What factor is missing?

    Réponse

    The 0.20 prior. Its striped branch weight is 0.20 × 0.70 = 0.14.

  9. Carte 9

    Question

    Only routes A and B are possible. Their priors are 0.4 and 0.6; P(chime | A) = 0.9 and P(chime | B) = 0.1. After a chime, what is P(A | chime)?

    Réponse

    6/7. A contributes 0.36; both routes contribute 0.36 + 0.06 = 0.42. Divide 0.36/0.42.

  10. Carte 10

    Question

    Only A and B exist. A learner divides A’s bell branch weight by itself and calls the result P(A | bell). What denominator term is missing?

    Réponse

    P(B)P(bell | B). The bell denominator must include B’s branch too, if B can produce a bell.

  11. Carte 11

    Question

    Only bins A and B exist. A’s share rises from 0.1 to 0.4 while P(glow | A) = 0.8 and P(glow | B) = 0.2 stay fixed. Does P(A | glow) rise or fall?

    Réponse

    It rises. The posterior changes from 4/13 to 8/11 because the prior for A increased with both likelihoods fixed.

  12. Carte 12

    Question

    Only sources A, B, and C exist. Their priors are 0.2, 0.3, 0.5; their beep rates are 0.5, 0.4, 0.2. What is P(beep)?

    Réponse

    0.32. Add every branch: 0.10 + 0.12 + 0.10.

  13. Carte 13

    Question

    For a source A, a learner uses P(A) × P(A) × P(glow | A) as its glow branch weight. What is the correction?

    Réponse

    Use the prior once. The branch weight is P(A) × P(glow | A).

  14. Carte 14

    Question

    Only sources A and B exist. A supplies 35% of items. What prior belongs on B’s branch?

    Réponse

    65%. The two source shares must add to 100%.

  15. Carte 15

    Question

    After a red result, the only three source-and-red weights are A: 0.12, B: 0.08, C: 0.20. What is P(A | red)?

    Réponse

    0.3. The total red weight is 0.40, so normalize A’s 0.12/0.40.

  16. Carte 16

    Question

    Only A and B are equally likely. Given each source, two new signals are independent. Their likelihoods are 0.8 and 0.6 under A, and 0.2 and 0.3 under B. After both signals, what is P(A | both)?

    Réponse

    8/9. Joint weights are 0.5 × 0.8 × 0.6 = 0.24 for A and 0.5 × 0.2 × 0.3 = 0.03 for B.

  17. Carte 17

    Question

    Only routes North and South exist. North is used 70% of the time, and P(late | North) = 10%. What is the North-and-late branch weight?

    Réponse

    7%. Multiply 70% × 10%; it is not the posterior P(North | late).

  18. Carte 18

    Question

    Sources A, B, C have priors 0.1, 0.3, 0.6 and signal likelihoods 0.9, 0.5, 0.4. Which source has the largest signal branch weight?

    Réponse

    C. The weights are 0.09, 0.15, and 0.24; A has the largest likelihood but not the largest contribution.

  19. Carte 19

    Question

    A model gives signal rates for sources A and B but says a third source may exist, without its share or signal rate. Can P(A | signal) be determined?

    Réponse

    No. The evidence total needs every possible source branch; an unmodelled source leaves the denominator incomplete.

  20. Carte 20

    Question

    Only sources Z and Y are possible. Their priors are 0.3 and 0.7; P(ring | Z) = 0.4 and P(ring | Y) = 0.1. After a ring, what is P(Z | ring)?

    Réponse

    12/19. The branch weights are 0.12 and 0.07; normalize Z’s weight by 0.19.

  21. Carte 21

    Question

    Only machines A and B produce tokens. Their shares are 0.6 and 0.4, and their stripe rates are 0.25 and 0.5. What is P(stripe)?

    Réponse

    0.35. Add 0.6 × 0.25 and 0.4 × 0.5, giving 0.15 + 0.20.

  22. Carte 22

    Question

    After a pulse, the only source-and-pulse weights are A: 0.10, B: 0.20, C: 0.30. What is P(A or B | pulse)?

    Réponse

    0.5. A and B contribute 0.30 of the total 0.60 pulse weight.

  23. Carte 23

    Question

    After one observation, P(A | first result) = 0.7. A genuinely new second result arrives. What prior for A enters the next conditional update?

    Réponse

    0.7. The first posterior becomes the starting probability for the new evidence; its likelihood still needs a model.

  24. Carte 24

    Question

    After a click, the only branch weights are A: 0.10 and B: 0.30. A learner says P(A | click) = 0.75. What is the corrected value?

    Réponse

    0.25. Normalize A’s weight, 0.10/(0.10 + 0.30); 0.75 belongs to B.

  25. Carte 25

    Question

    Only bags A and B are possible. Their priors are 0.2 and 0.8; P(star | A) = 0.75 and P(star | B) = 0.25. After seeing a star, what is P(A | star)?

    Réponse

    3/7. A’s star weight is 0.15 and total star weight is 0.15 + 0.20 = 0.35.

  26. Carte 26

    Question

    Only sources A, B, C have priors 0.25, 0.25, 0.5 and tick likelihoods 0.4, 0.2, 0.3. A proposed P(tick) is 0.15 because it omits C. What is the corrected P(tick)?

    Réponse

    0.30. Add C’s 0.5 × 0.3 = 0.15 to the A and B weights, 0.10 + 0.05.

  27. Carte 27

    Question

    Only A and B exist and their priors are known. P(ping | A) is known, but P(ping | B) is missing. Is P(A | ping) uniquely determined?

    Réponse

    No. B’s unknown ping contribution changes the denominator.

  28. Carte 28

    Question

    Only A and B make tiles. P(A) = 0.2, P(B) = 0.8, and P(star | A) = 0.7. A learner uses 0.8 × 0.7 for A’s star branch. What is A’s correct branch weight?

    Réponse

    0.14. A’s prior is 0.2, so multiply 0.2 × 0.7. The 0.8 prior belongs to B.

  29. Carte 29

    Question

    A learner wants P(not A | ring) but subtracts P(A | no ring) from 1. Which probability should be subtracted from 1?

    Réponse

    P(A | ring). Keep the observed condition fixed when taking the source complement.

  30. Carte 30

    Question

    After a stripe, the only three source-and-stripe weights are A: 0.06, B: 0.09, C: 0.15. What is P(C | stripe)?

    Réponse

    0.5. Divide C’s 0.15 by the total 0.30.

  31. Carte 31

    Question

    A sensor result appears twice because the screen refreshed; no second measurement occurred. Should a Bayes update use the same result twice?

    Réponse

    No. A duplicated display supplies no new evidence. Updating twice would count one observation twice.

  32. Carte 32

    Question

    Only A and B exist. P(A) = 0.1, P(B) = 0.9, P(chime | A) = 0.9, and P(chime | B) = 0.4. A learner chooses A as more likely after a chime because 0.9 > 0.4. Which source is actually more likely?

    Réponse

    B. The chime weights are A: 0.09 and B: 0.36; priors change the comparison.

  33. Carte 33

    Question

    Only boxes A and B exist. P(A) = 0.3, and both boxes yield a square token with probability 0.4. After seeing a square, what is P(A | square)?

    Réponse

    0.3. Equal likelihoods leave the prior unchanged: 0.3 × 0.4 / 0.4 = 0.3.

  34. Carte 34

    Question

    Three mutually exclusive sources cover every possibility. After observing a flash with positive probability, what must their three posterior probabilities sum to?

    Réponse

    1. Normalization divides every source-and-flash weight by the same positive total.

  35. Carte 35

    Question

    Two signals are independent overall, but their dependence given each hidden source is unknown. Can their source-specific likelihoods safely be multiplied?

    Réponse

    No. Multiplication for a source needs independence conditional on that source, or a supplied joint likelihood.

  36. Carte 36

    Question

    After a pulse, the only two source-and-pulse weights are 0.40 and 0.10. A learner reports the first source’s posterior as 0.40. What is it?

    Réponse

    0.8. Divide 0.40 by the total pulse weight 0.50; a branch weight is not yet a posterior.

  37. Carte 37

    Question

    Only trays A and B exist and are equally likely. Their dot rates are 0.6 and 0.3. After seeing a dot, what is P(A | dot)?

    Réponse

    2/3. The weights are 0.30 and 0.15; A contributes 0.30/0.45.

  38. Carte 38

    Question

    Only sensors A, B, C are possible. Their priors are 0.5, 0.3, 0.2 and ping likelihoods are 0.1, 0.4, 0.5. After a ping, what is P(C | ping)?

    Réponse

    10/27. C contributes 0.10; all sources contribute 0.05 + 0.12 + 0.10 = 0.27.

  39. Carte 39

    Question

    A model assigns an observed flash probability of zero across every possible source. Is P(A | flash) defined by the ordinary Bayes ratio?

    Réponse

    No. Its denominator P(flash) is zero, so the conditional probability is undefined in this discrete model.

  40. Carte 40

    Question

    Only sources A and B exist. A supplies 5% of items; P(flash | A) = 0.9 and P(flash | B) = 0.1. What is P(flash)?

    Réponse

    0.14. Add 0.05 × 0.9 and 0.95 × 0.1, giving 0.045 + 0.095.

  41. Carte 41

    Question

    Only A, B, C have priors 0.1, 0.4, 0.5 and mark likelihoods 0.9, 0.1, 0.1. Which source is most probable after a mark?

    Réponse

    A. Its weight 0.09 exceeds B’s 0.04 and C’s 0.05, despite A’s smallest prior.

  42. Carte 42

    Question

    Signals are independent given source A, but nothing is stated about their dependence given source B. Can both sources’ two-signal branch weights be built by multiplying separate likelihoods?

    Réponse

    No. The B branch still needs P(both signals | B) or conditional independence given B.

  43. Carte 43

    Question

    A proposed evidence total adds source A, source B, and a third label that includes every B item. What is the structural error?

    Réponse

    B is counted twice. The branches used in a total must be disjoint and cover all possible sources.

  44. Carte 44

    Question

    An observed click has branch weights 0.12 from source A and 0.18 from source B; A and B are the only sources. What is P(A | click)?

    Réponse

    0.4. Normalize A’s weight: 0.12/(0.12 + 0.18).

  45. Carte 45

    Question

    After a glow, the only source-and-glow weights are A: 0.08, B: 0.12, C: 0.20. What is P(not C | glow)?

    Réponse

    0.5. A and B contribute 0.20 of the 0.40 total; the complement is taken after conditioning on glow.

  46. Carte 46

    Question

    A classroom model assigns fixed bag shares and color rates. Does a posterior computed from it establish the source of a real token outside that model?

    Réponse

    No. The result is conditional on the stipulated shares, rates, and source list; a real setting needs checked assumptions and data.

  47. Carte 47

    Question

    Only sources A and B exist with priors 0.4 and 0.6. Supplied joint likelihoods for two signals are P(both | A) = 0.12 and P(both | B) = 0.24. What is P(A | both)?

    Réponse

    0.25. The weights are 0.4 × 0.12 = 0.048 and 0.6 × 0.24 = 0.144; divide 0.048 by 0.192.

  48. Carte 48

    Question

    Only sources A and B exist. P(A) = 0.4, P(light | A) = 0.8, and P(light | B) = 0.3. After no light, what is P(A | no light)?

    Réponse

    4/25. No-light weights are 0.4 × 0.2 = 0.08 and 0.6 × 0.7 = 0.42; divide 0.08 by 0.50.

Two small bags of teal and amber wooden tokens beside a bowl containing both colors on a wooden desk.

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Bayes’ Theorem Flashcards: Base Rates & Evidence Updates

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