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