Bayes’ Theorem Flashcards: Base Rates & Evidence Updates

Question 1

For a Bayes update over hidden sources, what must the listed sources cover?

Accepted Answer

Every possible source, exactly once. The sources must be exhaustive and mutually exclusive so their evidence branches can be added.

Question 2

Before seeing a result, a token comes from bag A with probability 0.3. What is 0.3 called in a Bayes update?

Accepted Answer

The prior for bag A. It is A’s source probability before the result is observed.

Question 3

A token from bag A is blue with probability 0.8. In a Bayes update after seeing blue, what role does 0.8 play?

Accepted Answer

The likelihood of blue given A. It is P(blue | A), not the chance that a blue token came from A.

Question 4

An observed signal can come from every possible source. What probability belongs in the denominator of a source posterior?

Accepted Answer

The total probability of the signal. Add its prior × likelihood branch weight across all possible sources.

Question 5

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?

Accepted Answer

0.24. Multiply 0.3 × 0.8; this is P(A and red), before normalization.

Question 6

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?

Accepted Answer

The condition was reversed. 0.8 is P(blue | bag); P(bag | blue) also needs source priors and the other bags’ blue rates.

Question 7

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)?

Accepted Answer

0.30. Add both marked branches: 0.25 × 0.6 + 0.75 × 0.2 = 0.15 + 0.15.

Question 8

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?

Accepted Answer

The 0.20 prior. Its striped branch weight is 0.20 × 0.70 = 0.14.

Question 9

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)?

Accepted Answer

6/7. A contributes 0.36; both routes contribute 0.36 + 0.06 = 0.42. Divide 0.36/0.42.

Question 10

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?

Accepted Answer

P(B)P(bell | B). The bell denominator must include B’s branch too, if B can produce a bell.

Question 11

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?

Accepted Answer

It rises. The posterior changes from 4/13 to 8/11 because the prior for A increased with both likelihoods fixed.

Question 12

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)?

Accepted Answer

0.32. Add every branch: 0.10 + 0.12 + 0.10.

Question 13

For a source A, a learner uses P(A) × P(A) × P(glow | A) as its glow branch weight. What is the correction?

Accepted Answer

Use the prior once. The branch weight is P(A) × P(glow | A).

Question 14

Only sources A and B exist. A supplies 35% of items. What prior belongs on B’s branch?

Accepted Answer

65%. The two source shares must add to 100%.

Question 15

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)?

Accepted Answer

0.3. The total red weight is 0.40, so normalize A’s 0.12/0.40.

Question 16

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)?

Accepted Answer

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.

Question 17

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?

Accepted Answer

7%. Multiply 70% × 10%; it is not the posterior P(North | late).

Question 18

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?

Accepted Answer

C. The weights are 0.09, 0.15, and 0.24; A has the largest likelihood but not the largest contribution.

Question 19

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?

Accepted Answer

No. The evidence total needs every possible source branch; an unmodelled source leaves the denominator incomplete.

Question 20

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)?

Accepted Answer

12/19. The branch weights are 0.12 and 0.07; normalize Z’s weight by 0.19.

Question 21

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)?

Accepted Answer

0.35. Add 0.6 × 0.25 and 0.4 × 0.5, giving 0.15 + 0.20.

Question 22

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)?

Accepted Answer

0.5. A and B contribute 0.30 of the total 0.60 pulse weight.

Question 23

After one observation, P(A | first result) = 0.7. A genuinely new second result arrives. What prior for A enters the next conditional update?

Accepted Answer

0.7. The first posterior becomes the starting probability for the new evidence; its likelihood still needs a model.

Question 24

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?

Accepted Answer

0.25. Normalize A’s weight, 0.10/(0.10 + 0.30); 0.75 belongs to B.

Question 25

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)?

Accepted Answer

3/7. A’s star weight is 0.15 and total star weight is 0.15 + 0.20 = 0.35.

Question 26

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)?

Accepted Answer

0.30. Add C’s 0.5 × 0.3 = 0.15 to the A and B weights, 0.10 + 0.05.

Question 27

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?

Accepted Answer

No. B’s unknown ping contribution changes the denominator.

Question 28

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?

Accepted Answer

0.14. A’s prior is 0.2, so multiply 0.2 × 0.7. The 0.8 prior belongs to B.

Question 29

A learner wants P(not A | ring) but subtracts P(A | no ring) from 1. Which probability should be subtracted from 1?

Accepted Answer

P(A | ring). Keep the observed condition fixed when taking the source complement.

Question 30

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)?

Accepted Answer

0.5. Divide C’s 0.15 by the total 0.30.

Question 31

A sensor result appears twice because the screen refreshed; no second measurement occurred. Should a Bayes update use the same result twice?

Accepted Answer

No. A duplicated display supplies no new evidence. Updating twice would count one observation twice.

Question 32

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?

Accepted Answer

B. The chime weights are A: 0.09 and B: 0.36; priors change the comparison.

Question 33

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)?

Accepted Answer

0.3. Equal likelihoods leave the prior unchanged: 0.3 × 0.4 / 0.4 = 0.3.

Question 34

Three mutually exclusive sources cover every possibility. After observing a flash with positive probability, what must their three posterior probabilities sum to?

Accepted Answer

1. Normalization divides every source-and-flash weight by the same positive total.

Question 35

Two signals are independent overall, but their dependence given each hidden source is unknown. Can their source-specific likelihoods safely be multiplied?

Accepted Answer

No. Multiplication for a source needs independence conditional on that source, or a supplied joint likelihood.

Question 36

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?

Accepted Answer

0.8. Divide 0.40 by the total pulse weight 0.50; a branch weight is not yet a posterior.

Question 37

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)?

Accepted Answer

2/3. The weights are 0.30 and 0.15; A contributes 0.30/0.45.

Question 38

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)?

Accepted Answer

10/27. C contributes 0.10; all sources contribute 0.05 + 0.12 + 0.10 = 0.27.

Question 39

A model assigns an observed flash probability of zero across every possible source. Is P(A | flash) defined by the ordinary Bayes ratio?

Accepted Answer

No. Its denominator P(flash) is zero, so the conditional probability is undefined in this discrete model.

Question 40

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)?

Accepted Answer

0.14. Add 0.05 × 0.9 and 0.95 × 0.1, giving 0.045 + 0.095.

Question 41

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?

Accepted Answer

A. Its weight 0.09 exceeds B’s 0.04 and C’s 0.05, despite A’s smallest prior.

Question 42

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?

Accepted Answer

No. The B branch still needs P(both signals | B) or conditional independence given B.

Question 43

A proposed evidence total adds source A, source B, and a third label that includes every B item. What is the structural error?

Accepted Answer

B is counted twice. The branches used in a total must be disjoint and cover all possible sources.

Question 44

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)?

Accepted Answer

0.4. Normalize A’s weight: 0.12/(0.12 + 0.18).

Question 45

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)?

Accepted Answer

0.5. A and B contribute 0.20 of the 0.40 total; the complement is taken after conditioning on glow.

Question 46

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?

Accepted Answer

No. The result is conditional on the stipulated shares, rates, and source list; a real setting needs checked assumptions and data.

Question 47

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)?

Accepted Answer

0.25. The weights are 0.4 × 0.12 = 0.048 and 0.6 × 0.24 = 0.144; divide 0.048 by 0.192.

Question 48

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)?

Accepted Answer

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