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