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.
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Κάρτα 1
Ερώτηση
For a Bayes update over hidden sources, what must the listed sources cover?
Απάντηση
Every possible source, exactly once. The sources must be exhaustive and mutually exclusive so their evidence branches can be added.
Κάρτα 2
Ερώτηση
Before seeing a result, a token comes from bag A with probability 0.3. What is 0.3 called in a Bayes update?
Απάντηση
The prior for bag A. It is A’s source probability before the result is observed.
Κάρτα 3
Ερώτηση
A token from bag A is blue with probability 0.8. In a Bayes update after seeing blue, what role does 0.8 play?
Απάντηση
The likelihood of blue given A. It is P(blue | A), not the chance that a blue token came from A.
Κάρτα 4
Ερώτηση
An observed signal can come from every possible source. What probability belongs in the denominator of a source posterior?
Απάντηση
The total probability of the signal. Add its prior × likelihood branch weight across all possible sources.
Κάρτα 5
Ερώτηση
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?
Απάντηση
0.24. Multiply 0.3 × 0.8; this is P(A and red), before normalization.
Κάρτα 6
Ερώτηση
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?
Απάντηση
The condition was reversed. 0.8 is P(blue | bag); P(bag | blue) also needs source priors and the other bags’ blue rates.
Κάρτα 7
Ερώτηση
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)?
Απάντηση
0.30. Add both marked branches: 0.25 × 0.6 + 0.75 × 0.2 = 0.15 + 0.15.
Κάρτα 8
Ερώτηση
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?
Απάντηση
The 0.20 prior. Its striped branch weight is 0.20 × 0.70 = 0.14.
Κάρτα 9
Ερώτηση
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)?
Απάντηση
6/7. A contributes 0.36; both routes contribute 0.36 + 0.06 = 0.42. Divide 0.36/0.42.
Κάρτα 10
Ερώτηση
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?
Απάντηση
P(B)P(bell | B). The bell denominator must include B’s branch too, if B can produce a bell.
Κάρτα 11
Ερώτηση
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?
Απάντηση
It rises. The posterior changes from 4/13 to 8/11 because the prior for A increased with both likelihoods fixed.
Κάρτα 12
Ερώτηση
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)?
Απάντηση
0.32. Add every branch: 0.10 + 0.12 + 0.10.
Κάρτα 13
Ερώτηση
For a source A, a learner uses P(A) × P(A) × P(glow | A) as its glow branch weight. What is the correction?
Απάντηση
Use the prior once. The branch weight is P(A) × P(glow | A).
Κάρτα 14
Ερώτηση
Only sources A and B exist. A supplies 35% of items. What prior belongs on B’s branch?
Απάντηση
65%. The two source shares must add to 100%.
Κάρτα 15
Ερώτηση
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)?
Απάντηση
0.3. The total red weight is 0.40, so normalize A’s 0.12/0.40.
Κάρτα 16
Ερώτηση
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)?
Απάντηση
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.
Κάρτα 17
Ερώτηση
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?
Απάντηση
7%. Multiply 70% × 10%; it is not the posterior P(North | late).
Κάρτα 18
Ερώτηση
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?
Απάντηση
C. The weights are 0.09, 0.15, and 0.24; A has the largest likelihood but not the largest contribution.
Κάρτα 19
Ερώτηση
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?
Απάντηση
No. The evidence total needs every possible source branch; an unmodelled source leaves the denominator incomplete.
Κάρτα 20
Ερώτηση
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)?
Απάντηση
12/19. The branch weights are 0.12 and 0.07; normalize Z’s weight by 0.19.
Κάρτα 21
Ερώτηση
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)?
Απάντηση
0.35. Add 0.6 × 0.25 and 0.4 × 0.5, giving 0.15 + 0.20.
Κάρτα 22
Ερώτηση
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)?
Απάντηση
0.5. A and B contribute 0.30 of the total 0.60 pulse weight.
Κάρτα 23
Ερώτηση
After one observation, P(A | first result) = 0.7. A genuinely new second result arrives. What prior for A enters the next conditional update?
Απάντηση
0.7. The first posterior becomes the starting probability for the new evidence; its likelihood still needs a model.
Κάρτα 24
Ερώτηση
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?
Απάντηση
0.25. Normalize A’s weight, 0.10/(0.10 + 0.30); 0.75 belongs to B.
Κάρτα 25
Ερώτηση
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)?
Απάντηση
3/7. A’s star weight is 0.15 and total star weight is 0.15 + 0.20 = 0.35.
Κάρτα 26
Ερώτηση
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)?
Απάντηση
0.30. Add C’s 0.5 × 0.3 = 0.15 to the A and B weights, 0.10 + 0.05.
Κάρτα 27
Ερώτηση
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?
Απάντηση
No. B’s unknown ping contribution changes the denominator.
Κάρτα 28
Ερώτηση
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?
Απάντηση
0.14. A’s prior is 0.2, so multiply 0.2 × 0.7. The 0.8 prior belongs to B.
Κάρτα 29
Ερώτηση
A learner wants P(not A | ring) but subtracts P(A | no ring) from 1. Which probability should be subtracted from 1?
Απάντηση
P(A | ring). Keep the observed condition fixed when taking the source complement.
Κάρτα 30
Ερώτηση
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)?
Απάντηση
0.5. Divide C’s 0.15 by the total 0.30.
Κάρτα 31
Ερώτηση
A sensor result appears twice because the screen refreshed; no second measurement occurred. Should a Bayes update use the same result twice?
Απάντηση
No. A duplicated display supplies no new evidence. Updating twice would count one observation twice.
Κάρτα 32
Ερώτηση
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?
Απάντηση
B. The chime weights are A: 0.09 and B: 0.36; priors change the comparison.
Κάρτα 33
Ερώτηση
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)?
Απάντηση
0.3. Equal likelihoods leave the prior unchanged: 0.3 × 0.4 / 0.4 = 0.3.
Κάρτα 34
Ερώτηση
Three mutually exclusive sources cover every possibility. After observing a flash with positive probability, what must their three posterior probabilities sum to?
Απάντηση
1. Normalization divides every source-and-flash weight by the same positive total.
Κάρτα 35
Ερώτηση
Two signals are independent overall, but their dependence given each hidden source is unknown. Can their source-specific likelihoods safely be multiplied?
Απάντηση
No. Multiplication for a source needs independence conditional on that source, or a supplied joint likelihood.
Κάρτα 36
Ερώτηση
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?
Απάντηση
0.8. Divide 0.40 by the total pulse weight 0.50; a branch weight is not yet a posterior.
Κάρτα 37
Ερώτηση
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)?
Απάντηση
2/3. The weights are 0.30 and 0.15; A contributes 0.30/0.45.
Κάρτα 38
Ερώτηση
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)?
Απάντηση
10/27. C contributes 0.10; all sources contribute 0.05 + 0.12 + 0.10 = 0.27.
Κάρτα 39
Ερώτηση
A model assigns an observed flash probability of zero across every possible source. Is P(A | flash) defined by the ordinary Bayes ratio?
Απάντηση
No. Its denominator P(flash) is zero, so the conditional probability is undefined in this discrete model.
Κάρτα 40
Ερώτηση
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)?
Απάντηση
0.14. Add 0.05 × 0.9 and 0.95 × 0.1, giving 0.045 + 0.095.
Κάρτα 41
Ερώτηση
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?
Απάντηση
A. Its weight 0.09 exceeds B’s 0.04 and C’s 0.05, despite A’s smallest prior.
Κάρτα 42
Ερώτηση
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?
Απάντηση
No. The B branch still needs P(both signals | B) or conditional independence given B.
Κάρτα 43
Ερώτηση
A proposed evidence total adds source A, source B, and a third label that includes every B item. What is the structural error?
Απάντηση
B is counted twice. The branches used in a total must be disjoint and cover all possible sources.
Κάρτα 44
Ερώτηση
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)?
Απάντηση
0.4. Normalize A’s weight: 0.12/(0.12 + 0.18).
Κάρτα 45
Ερώτηση
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)?
Απάντηση
0.5. A and B contribute 0.20 of the 0.40 total; the complement is taken after conditioning on glow.
Κάρτα 46
Ερώτηση
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?
Απάντηση
No. The result is conditional on the stipulated shares, rates, and source list; a real setting needs checked assumptions and data.
Κάρτα 47
Ερώτηση
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)?
Απάντηση
0.25. The weights are 0.4 × 0.12 = 0.048 and 0.6 × 0.24 = 0.144; divide 0.048 by 0.192.
Κάρτα 48
Ερώτηση
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)?
Απάντηση
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.
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Bayes’ Theorem Flashcards: Base Rates & Evidence Updates
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