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