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Incoming links: Conditional probability
The probability of two black balls is the complement of the other two possibilities:The probability of equal colours is . Thus the required conditional probability isThis is defined when the conditioning event has positive probability. If with one ball of each colour, that event is impossible and the conditional probability is undefined.
Write . Successive conditional probabilities account for the changing bag under sampling without replacement:On this event no equality of the two colour counts has occurred, so the game's stopping rule does not interrupt these three draws.
Let and denote the first and second labels. Sampling with replacement restores the bag before the second draw, so conditioning on only specifies the threshold. For , there are qualifying green balls and qualifying red balls. For , every green ball and the first red balls qualify. Hence the requested conditional probability isThe formulas agree at and give one at . Labels outside the possible first-draw range do not define a conditioning event of positive probability.
There are two equally likely balls bearing label eight, one of each colour. The conditional probability that the removed ball was green is therefore . By the law of total probability,Both possible first colours must be included because the observed label alone does not determine the colour.
Only a green ball carries label four. After it is removed, nine of the remaining nineteen balls are green. The conditional probability is
Treat each observed arrival label as a constraint on an unobserved walking duration. Departures are treated as fixed design information. Writing for the minutes available until work starts, the derived late-label thresholds have frequencies ; the early-label thresholds have frequencies . These are censoring thresholds, not measured journey times. A late label means , and an early label means ; equality has probability zero under the continuous model.
The paper specifies no prior distribution. One explicit proper model iswith independent priors; all times are in minutes. These ranges are a transparent modelling choice, not information deduced from the observations. In particular, this is a prior on the standard deviation , not a uniform prior on the variance . The normal distribution permits negative durations; it is the mathematical approximation requested here, rather than a claim that such durations are physically possible. A truncated model would be a different model.
The following Python implements hard conditioning by rejection sampling.
None marks a rejected trace; an accepted Boolean answers whether all five future arrivals are late. The same parameters are shared across past and future days.import math
import random
LATE = {20: 1, 21: 3, 25: 3, 27: 1, 32: 2}
EARLY = {29: 1, 34: 1, 36: 3, 37: 1, 38: 1, 39: 1, 43: 1, 44: 1}
RECORDS = [(c, True) for c, n in LATE.items() for _ in range(n)]
RECORDS += [(c, False) for c, n in EARLY.items() for _ in range(n)]
def journey_trace(rng):
mu = rng.uniform(0.0, 60.0)
sigma = rng.uniform(1.0, 20.0)
# Twenty latent past durations and five future durations.
past = [rng.gauss(mu, sigma) for _ in RECORDS]
future = [rng.gauss(mu, sigma) for _ in range(5)]
if not all(t > c if late else t < c
for t, (c, late) in zip(past, RECORDS)):
return None
return all(t > 30.0 for t in future)
def accepted_average(trace, n, rng):
if n <= 0:
raise ValueError("n must be positive")
successes = accepted = 0
while accepted < n:
result = trace(rng)
if result is not None:
accepted += 1
successes += int(result)
return successes / nFor example,
accepted_average(journey_trace, n, random.Random(2021)) estimates the desired conditional probability from accepted traces. Accepting False is essential: treating a false answer as a rejected trace would condition on the desired conclusion. The model deliberately emphasizes clear semantics; rejection can be inefficient because all twenty past-label constraints must hold together.For any chosen proper prior density on with positive evidence, Bayes' theorem givesThe prior density comes from the first two random draws. The likelihood is the probability of satisfying all observed-label constraints after integrating the past duration noises. The factor is the probability of the five-day outcome conditional on the shared parameters. The denominator is the probability of acceptance, which converts the numerator's joint probability into the required conditional probability.
For the independent uniform priors in part (a), the constant cancels and both integrals reduce to , . A prior on the variance would require transforming the density: this chosen prior has density for conditional on . One must not silently replace it by a uniform prior on .
There is no unique numerical probability until the prior distribution is chosen. As a check for the stated uniform priors, independent Gaussian quadrature calculations with 256 and 512 nodes per coordinate agree at the shown precision:The prior-predictive probability of all observed labels is approximately , explaining the rejection cost. These numbers are illustrations of the explicit model, not additional observed data.
A subtle point is that the five future days are independent only conditional on . Integrating shared parameter uncertainty gives , not . The latter is about for these priors and understates the five-day probability. Jensen's inequality gives , while gives . For two future late-arrival indicator functions, their posterior covariance is . This is the posterior predictive repeated-success identity.
Posterior uncertainty and the five-day predictive probability for the explicitly chosen uniform priors. The contour is cropped to the parameter region carrying the visible mass; integration uses the full prior rectangle.
Details. 
Use the standard repeated game convention that the first round's joint action is observed before round two. The following original game tree represents simultaneous moves by making row move first in each round while hiding that current move from column. Terminal labels are total payoffs over both rounds, ordered row then column.
The full two-round extensive-form game, with all sixteen terminal payoff pairs and five column information sets
. Details. 
Formally, round one starts at a row decision with actions . The two ensuing column nodes form one information set , with available actions ; column cannot tell which current row action occurred. The four completed first-round histories are . At each , round two begins at a separate row node. Its two successors, after row's current or , form a column information set . Distinct have distinct information sets: both players know the whole first-round outcome. There are sixteen terminal histories , whose total payoff is , as displayed in the tree.
The row player's five information sets are the initial singleton and the four singleton round-two nodes, one per . The column player's five information sets are and the four two-node sets . Thus each player must specify one binary choice at each of five information sets, including histories that its own first choice might prevent:Counting only the four possible two-round action sequences would miss the distinct contingent responses to the opponent's first action.
For a concrete pair of complete pure strategies, order the second-round histories as and useThe first entry is the initial action, and the four remaining entries prescribe the action after each first-round history. This induces the unique path in round one followed by in round two. Hence the total payoff isThe off-path entries are part of each pure strategy, even though this pair does not reach them.
For randomization, a mixed strategy is a probability distribution over the complete plans, with free probabilities in a general distribution. A behavioural strategy instead gives one binary-action probability at each of five information sets, requiring five parameters. This game has perfect recall: a player never forgets a previous own action or a first-round observation, and the only concealed action is the opponent's current simultaneous choice. Consequently neither kind of randomization has a greater attainable-payoff advantage: Kuhn's theorem gives realization-equivalent strategies against every fixed opposing strategy.[ref]
Extensive games and the problem of information by Harold W. Kuhn. original article, pp. 193–216, equivalence under perfect recall
The equivalence can be verified directly here. Given a behavioural strategy, sample all five contingent actions independently before play; the resulting product probability distribution over complete plans is an equivalent mixed strategy. Conversely, let be a mixed strategy for player . Use its marginal probability for the initial action. At history containing own first action , choose second action with conditional probabilitywhenever the denominator is positive; if it is zero, choose arbitrarily at that unreachable history. The opponent's first-round action is simultaneous and cannot depend on the sampled hidden plan beyond the chosen first action. Thus conditioning on preserves exactly these conditional probabilities of actions, and multiplying the first- and second-round factors reproduces each terminal probability. Correlations between prescriptions at mutually exclusive second-round histories have no further observable effect. Behavioural strategies offer the more compact representation, while mixed strategies describe a single random choice of a full plan. Imperfect information here does not mean imperfect recall.
The covering-law model explains a phenomenon by subsuming it under scientific laws and specified circumstances. Its deductive-nomological model requires the event to follow by valid deduction, with a law used essentially, empirical content and true premises. For example, a dynamical law and appropriate initial conditions can explain a body's later trajectory.[ref]
Studies in the Logic of Explanation by Carl Hempel and Paul Oppenheim. section 3: logical and empirical adequacy conditions
This captures an important feature of many physical explanations: the event is shown to fit a wider, independently supported pattern. It makes assumptions explicit and connects explanation with prediction. However, derivability alone does not ensure the right explanatory direction. With a pole, a known solar elevation and a measured shadow, geometry lets us calculate either height from shadow or shadow from height. The explanatory asymmetry remains: under the ordinary causal setup, the pole and illumination explain the shadow, rather than its measured length causing the pole's height.
Irrelevant information also exposes a weakness. From a general law plus an unnecessary fact one may still derive the result; logical validity does not show that the added fact helps explain it. Conversely, many accepted explanations are statistical or mechanistic rather than deductions from exceptionless universal laws. Extending the account to the inductive-statistical model helps, but a high conditional probability does not by itself establish causal relevance.
The covering-law model is thus a useful account of one important explanatory structure, not a complete criterion of scientific explanation. A stronger account must consider which assumptions make a difference, what mechanism operates and why the explanatory direction is appropriate. The model's virtues survive when it is treated as a component of explanation rather than an exhaustive definition.
The covering-law model captures one important feature of scientific explanation: a result should follow from general principles together with the relevant circumstances. In its deductive-nomological model, laws and conditions entail the phenomenon, the laws are needed in the derivation, and the premises must be true and empirically substantive. That is stronger than merely redescribing the event.[ref]
Studies in the Logic of Explanation by Carl Hempel and Paul Oppenheim. section 3, conditions R1–R4
For example, the ideal gas law, a specified amount of gas and a fixed-volume condition explain why changing its temperature changes its pressure in the model's regime. The derivation identifies a systematic dependence rather than an isolated coincidence. It also makes assumptions visible: a calculation using a law outside its domain is not repaired simply by being logically valid.
Nevertheless, valid subsumption does not by itself determine explanatory direction. A vertical pole of height , a solar elevation and a shadow of length satisfy under suitable geometric assumptions. The same relation allows one to deduce the height from the shadow. In an ordinary intervention on the pole, changing its height changes its shadow; shortening the shadow with a screen does not shorten the pole. The reverse calculation can be excellent evidence for the height without providing the same causal explanation. This is explanatory asymmetry.
Relevance is another difficulty. One can append an irrelevant fact to a sound derivation without invalidating the deduction, but the added fact need not explain anything. Requiring an essential law prevents some vacuous derivations, yet still does not identify which features make a physical difference. A useful explanation should reveal the dependence the question is asking about, not merely contain enough true premises to reach its conclusion.
Nor are all explanations deterministic. A particular radioactive decay can have a good account in terms of a physical decay mechanism and a probability distribution without being entailed by initial conditions. An inductive-statistical model extends the covering-law idea, but requiring a high probability would exclude the explanation of some rare outcomes. Conversely, a high conditional probability produced by irrelevant correlations need not explain a mechanism.
The model is therefore a valuable discipline for organizing some explanations, rather than a sufficient universal analysis. It clarifies derivational structure and testable commitments. An adequate broader account must also address causal relevance, the question's contrast, statistical events and explanations by constraints or unification that are not straightforward causal histories.
A Bayesian confirmation account can dissolve the apparent conflict between logical equivalence and ordinary judgements about relevant evidence. It does so by rejecting an unrestricted positive-instance rule and specifying how the evidence was sampled. Bayes' theorem alone does not guarantee that every white shoe is irrelevant, or that every black raven provides strong support.
Let assert that every raven is black. Its contrapositive is logically equivalent to , so the two descriptions must receive the same conditional probability. The raven paradox arises if every positive instance of a universal statement automatically confirms it: a white nonraven then confirms the contrapositive and hence . The disputed step is that unrestricted confirmation rule, not the logical equivalence.[ref]
Studies in the Logic of Confirmation I by Carl Hempel. section 5: equivalence and the paradoxes of instance confirmation
For background , set , and . For and positive evidence probability,Thus it is the comparison of likelihoods, including the sampling protocol and alternatives, that determines support. Merely rewriting cannot change it.
The finite-population model of raven confirmation makes the intuitive difference quantitative. Fix ravens and nonblack objects. Compare an all-black-ravens hypothesis with the alternative that exactly one raven is nonblack. A uniform draw among nonblack objects that finds a nonraven has likelihood ratio . A uniform draw among ravens that finds a black one has likelihood ratio . If , the former is much nearer one. With equal prior probabilities, and give posterior probabilities approximately and . A tiny amount of confirmation need have no practical significance.
This example is not a theorem about every possible background. Inspecting an object already known to be a white shoe adds no relevant information about its being a nonraven. Other alternatives or selection mechanisms change the likelihoods and can remove or reverse the support. Conditioning only on the observed colour while silently changing how the object was selected would compare different experiments.[ref]
How Bayesian Confirmation Theory Handles the Paradox of the Ravens by Branden Fitelson and James Hawthorne. Introduction and the treatment of background-sensitive confirmation; the finite model here is stated separately
The resolution is therefore qualified. A coherent probability model preserves equivalence and explains why superficially parallel observations need carry radically different evidence. It does not select a unique prior or sampling model without scientific judgement. Those remaining modelling questions are limitations of an application, rather than a contradiction in Bayesian inference.
The covering-law model explains an event by showing how it falls under general laws in specified circumstances. It captures a major form of scientific understanding, but entailment and predictability do not by themselves capture explanatory relevance or direction.
In the deductive-nomological model, let be laws, relevant circumstances, and the event description. The explanans is ; the explanandum is . The argument has the formThe deduction must be valid, at least one law must be essential to it, the explanans must have empirical content, and its premises must be true for a sound explanation. Merely writing a valid argument from arbitrary premises is insufficient. Carl Hempel and Paul Oppenheim also identify a structural affinity between explanation and prediction: supplied beforehand, the same premises could in principle support the prediction.[ref]
Studies in the Logic of Explanation by Carl Hempel and Paul Oppenheim. section 3: deductive schema, adequacy conditions and the explanation/prediction relationship
For an ideal simple pendulum at small amplitude, Newtonian mechanics and the specified constraints yieldHere is the pendulum's length, the local gravitational acceleration, and its oscillation period. The deduction connects an observed regularity to more general laws and explains why changing length changes the oscillation period. The approximation must be stated: a finite-amplitude, dissipative simple pendulum does not exactly satisfy these premises. A useful model-based explanation can be approximately correct even when the ideal premises are not literally true.
The model's appeal is that it requires more than an evocative story. It displays assumptions and shows why they constrain the phenomenon. It can also explain laws through wider laws: geometrical optics can recover a refraction law from suitable descriptions of wave propagation. A principle theory can explain by constraints without supplying a detailed constructive theory of every constituent process. This is a strength of the account's generality.
The first difficulty is asymmetry. For a vertical pole of vertical length on level ground and the Sun's elevation angle , geometrical optics gives the shadow's lengthWith suitable premises, either deduction is valid. Yet the pole and illumination explain the shadow, whereas measuring the shadow ordinarily allows inference to the pole's vertical length without explaining why the pole has that vertical length. The same equation supports both calculations. Its logical form does not encode the relevant direction of dependence. Context can change the question, but the distinction between explanation and evidential reconstruction remains.
The second difficulty is irrelevance. Adding the premise that a laboratory wall is blue to a correct simple pendulum derivation preserves validity and the essential use of physical laws. The extra premise contributes nothing to explaining the oscillation period. More elaborate examples show the same problem when irrelevant factors are hidden inside statistical regularities. The explanatory account needs a way to identify what matters under relevant changes, not just what accompanies a valid derivation.
The inductive-statistical model extends the programme to explanations using statistical laws and high conditional probability, relative to appropriate available information. This is an extension beyond the deductive focus of the 1948 article. High probability is nevertheless not necessary for every explanation. Radioactive decay can explain an infrequent detector event through a physical process and its nuclear decay constant, although that particular event in a short interval was unlikely. Nor does a high conditional probability automatically establish explanatory relevance: a reliable indicator may be an effect of the outcome or share another cause with it.
A better assessment therefore retains law-based derivations as one explanatory resource while adding dependence, relevance and the question being asked. Mechanistic explanations identify productive processes; accounts based on interventions ask which changes would alter the outcome. Neither proposal needs to force every mathematical or constraint-based explanation into a causal story. The covering-law model illuminates how laws make phenomena intelligible, but it is too broad as a sufficient test and too restrictive as a universal necessary form.
Past exam of the physics A course of the University of Cambridge 2022 Paper 1 Section A 4 Solution 2026-10-18
Let denote an incident muon and a detector signal. The selected sample is conditioned on . The law of total probability givesBy Bayes' theorem, the contamination, namely the conditional probability that a selected particle is not a muon, isFor example, per incident particles the expected selected counts are true muons and other particles. The denominator is the selected particles; the rejected muons are outside this sample.
Each is a bounded Borel measurable function of , including the specified value at . Therefore is an adapted process for and consists of integrable random variables.
Fix an atom of a sigma-algebra in with positive probability, corresponding to the dyadic interval , with . By the uniform distribution, each half has conditional probability . The conditional expectation of on that atom is thereforeThe middle values cancel. On the zero-probability atom the conditional expectation can be chosen as zero, agreeing with our version of . Henceso is a martingale. This is the dyadic slope martingale associated with the Lipschitz function .
Put . Before , all three players are active. Each unordered pair is selected with probability , and each direction of transfer then has conditional probability . Given the current holdings , a transfer within the first pair has mean productFor transfers within the other two pairs the analogous values are and . Averaging over pairs givesAt and after elimination the product is zero, since an eliminated player never re-enters. In particular the stopped process has incrementThe indicator function is -measurable, and the expression in parentheses has zero conditional expectation on its support. The stopped process is adapted and integrable: at fixed , its absolute value is bounded by . HenceThis is the product martingale for three-player token exchange.
The coefficient of the momentum eigenstate is . The Born rule therefore gives . Conditional on this outcome, the projective measurement prepares the post-measurement state , equivalent to up to global phase.
Since , the subsequent energy measurement has conditional probability one of giving the specified energy eigenvalue. The probability for the entire sequence is consequentlyAny intervening unitary time evolution under the same Hamiltonian operator preserves this energy eigenstate, up to global phase.
Continue with the initial quantum state from part (ii), and let . Conditional on every preceding outcome being , the post-measurement state after each ideal projective measurement is again , up to global phase. Part (ii) therefore gives the same conditional probability for each successive outcome. Multiplying these conditional probabilities yields the repeated projective-measurement survival probabilityThis multiplication uses the quantum state reset at each successful projective measurement; the outcomes need not be assumed statistically independent.
For fixed , put . The Taylor expansion gives the quadratic short-time quantum survival probability. For sufficiently large , taking logarithms givesHere the remainder uses Big O notation. Exponentiating provesThis is the quantum Zeno effect: increasingly frequent ideal projective measurements inhibit the transition away from the initial quantum state. The departure probability in a short interval is quadratic in its duration, so intervals of duration produce a total leading departure probability proportional to . Without intervening measurements, the return probability at time would be . The result concerns obtaining at every measurement in the ideal limit of instantaneous measurements with unitary time evolution between them.
Integrating the envelope shows . One trial of rejection sampling draws and an independent from the uniform distribution on , and accepts if . Repeat with fresh independent random variables after each rejection. Ratios need only be evaluated where ; the envelope makes the target zero wherever .
For any measurable set , integration of the conditional probability of acceptance givesTaking the whole domain gives acceptance probability , and dividing gives . Fresh trials therefore supply independent and identically distributed random variables with target probability density function . The corresponding Monte Carlo estimator isThe proposal count up to and including one acceptance has the geometric distributionFor example, the tail-sum formula gives . Thus the expected cost is proposal draws per accepted target draw.
For a measurable set , (i) gives the joint acceptance probabilityDivide by from (ii). The resulting conditional probability isThus every accepted value has target probability density function . Repeating rejected trials with fresh independent random variables does not change this law: the contribution from acceptance on trial is multiplied by , and summing that geometric series gives the same conditional law. Independent repetitions produce independent target draws.