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