Relevance and unreasonable rage

There may be occasions where a judge may rule that evidence initially thought to be relevant to an issue must be ignored. The Supreme Court of Canada has touched on this in the context of provocation: R v Tran [2010] SCC 58 (26 November 2010):

"[41] ... the evidence must be reasonably capable of supporting the inferences necessary to make out the defence before there is an air of reality to the defence (Fontaine 2004 SCC 27 (CanLII), 2004 SCC 27, [2004] 1 S.C.R. 702 at para. 56; R. v. Reddick, 1991 CanLII 106 (S.C.C.), [1991] 1 S.C.R. 1086, at p. 1088, citing Pappajohn v. The Queen, 1980 CanLII 13 (S.C.C.), [1980] 2 S.C.R. 120, at p. 133). ... ."

Tran is an example of the evidence being insufficient to establish a necessary element (here, of a defence), but I discuss it because it obliquely invites consideration of the admissibility/insufficiency boundary.

The "air of reality" requirement is really a relevance requirement.

To be relevant, evidence must be reasonably capable of supporting the fact it seeks to prove: Bain v R [2009] NZSC 16 per Elias CJ and Blanchard J at [43] (adopting R v Thomas [1970] VR 674 at p 679), per Wilson J at [91], agreeing, and adding that the threshold is "very low").

Judges may - unless there has been a pre-trial hearing - have to rule on admissibility when a party seeks to adduce the evidence, and that may be when there is only a narrow context. But on the "air of reality" test, the judge can take a wider perspective and place the challenged evidence in the context of all the other evidence in the case.

The danger is that, in a jury trial, the judge will usurp the jury's function by removing an issue (often, but not necessarily, a defence) from their consideration. The law of provocation, surveyed in Tran, brings out this difficulty, which reflects the sometimes controversial difference between questions of law and questions of fact.

The trial was by judge alone, and the judge held that the prosecution had failed to exclude the partial defence of provocation. The Court of Appeal of Alberta held that the offender should have been convicted of murder, not manslaughter. The Supreme Court dismissed the offender's appeal. Broadly, the offender could not have been acting under provocation because his estranged wife's adultery was known to him before he entered her home and it could not have been an "insult" (s 232 Criminal Code) that caused him to lose his self-control.

So the trial judge must have thought it was reasonably possible that the offender had lost his self-control (a question of fact), while the appellate courts did not (because of a requirement of law: the facts did not establish an "insult" although if they had, that would have been a question of fact under s 232). There is an objective element to the loss of self control:

"[29] ... while one spouse undoubtedly has a legal right to leave his or her partner, in some circumstances the means by which that spouse communicates this decision may amount in fact to an "insult", within the ordinary meaning of the word. However, to be recognized at law, the insult must be of sufficient gravity to cause a loss of self-control, as objectively determined. The fact that the victim has the "legal right", in the broad sense of the term, to leave the relationship is an important consideration in the assessment of this objective standard."

Tran was not so much an "air of reality" case, as a case of loss of self control being outside the scope of the partial defence of provocation because it was unreasonable. But to avoid the hideous spectacle of an appellate court acting like a jury, it is more diplomatic to pretend that the evidence was irrelevant.

Power to disclose

Does voluntary disclosure to the police by an electricity supply company of the defendant's power usage breach a right to privacy?

The Supreme Court of Canada has held, 7-2, that no such breach occurred in R v Gomboc [2010] SCC 55. The defendant was charged with cultivation of cannabis and his electricity usage was one aspect of the case against him. Four of the majority judges (Deschamps, Charron, Rothstein and Cromwell JJ) held that information about electricity use in a private dwelling was not about "intimate or core personal activities" of the occupants and did not carry a reasonable expectation of privacy. Core biographical data was not revealed. A relevant circumstance was that customers could expressly request that the supply company keep confidential the details about electricity usage, but here the defendant had not done that. This latter point was stressed by the concurring majority judges, Binnie, LeBel and Abella JJ.

Dissenting, McLaughlin CJ and Fish J held that there was a reasonable expectation of privacy in the circumstances of this case. This was because of the degree of intrusiveness was high: the accurate measuring of electricity usage supported inferences of criminal activity that was private information of use to the police. This form of search should only be permitted if the police could alternatively have obtained a search warrant. Customers could not be expected to be aware of the complex regulations which permitted the supply company to pass on information to the police, and the regulatory scheme was not intended to authorise the company to act as an agent for the police by spying on its customers.

Offenders often bypass their electricity meters, and are consequently charged with theft of electricity in addition to cultivation of cannabis. Police use of power consumption information in support of applications for search warrants may already be regarded as a good investigative technique; in R v Thompson [2001] 1 NZLR 129, (2000) 18 CRNZ 401 (CA) it does not appear to have occurred to anyone that this might have been objectionable.

Relevance, probative value, and Bayesian reasoning

Peter Tillers has drawn everyone's attention to an interesting discussion of relevance, "Bayesian Wars Redivivus – An Exchange" in International Commentary on Evidence, Vol 8, Issue 1, Article 1 (2010).

Relevance

Definitions of relevance can appear to be inappropriately restrictive. In New Zealand we have an example in s 7 of the Evidence Act 2006 (compare rule 403 of the Federal Rules of Evidence 2010). Critically, subsection (3) defines relevant evidence:

"Evidence is relevant in a proceeding if it has a tendency to prove or disprove anything that is of consequence to the determination of the proceeding."

Inevitably, however, there are many facts in a case that are not in dispute but that are relied on to give context to the relevant facts. These contextual matters are often called narrative. Strictly speaking, they do not in themselves tend to prove anything, but they help to explain why other evidence does have the necessary tendency. A generous reading of s 7(3) is needed: it could encompass narrative evidence if "evidence" is understood as including the evidence that explains the "tendency to prove or disprove".

Neutralising the opponent's evidence

Sometimes, each party may rely on the same narrative evidence but will invite a different inference from it. A defence stratagem is to neutralise prosecution evidence by showing that it is consistent with innocence, and this applies to narrative evidence too.

This leads to a paradox: if a narrative fact is not in dispute, but is equally consistent with guilt as with innocence, it is needed by both parties even though its tendency to prove a matter in issue is neutral; it is relevant in ways that cancel each other out. Does neutralisation destroy relevance?

This paradox disappears if s 7(3) means that equal and opposite tendencies are still tendencies, because each must be considered separately.

A problem for Bayesian logic?

The tendency requirement for relevance says nothing about the strength of that tendency. The strength of the tendency of relevant evidence to prove a matter in issue is called the probative value of that evidence. In Bayesian logic probative value is expressed as a likelihood ratio. Essentially this is a way of asking (for prosecution evidence) how much more consistent is the evidence with guilt than it is with innocence.

In some ways it is unfortunate that the adjective Bayesian has attached to this thought process, because it is an ordinary and natural way of addressing the question of the strength of the probative value of evidence. There is not even anything necessarily mathematical about it, as strengths and likelihoods can be assessed without numbers.

Much of the argument in the paper cited above is concerned with how a likelihood ratio is to deal with common reliance on the evidence without rendering it irrelevant. I think there is some crossing of the wires here: probative value is treated in this discussion as if it was relevance. This error is introduced by Ronald Allen at p 10 of the exchange. Roger Park tries to correct it at p 11, but David Kaye thinks it makes a different point (p 11). Ronald Allen emphasises his assertion of a problem that a likelihood ratio of one makes for relevance at p 12, David Kaye discusses the probative values in reply (p 12), and Ronald Allen's rejoinder (p 13) corrects the tendency to think that changing probative value changes relevance, while seeming to suggest that it was David Kaye who said that if both sides rely on the evidence it is not relevant at all, when really it was Ronald Allen himself who suggested that this is implicit in Bayesian reasoning. Ronald Allen thinks it is difficult to determine relevancy until all the evidence has been heard (p 13) – but I think that is because he confuses relevancy with probative value. Samuel Gross seems to agree that relevance cannot be assessed without the other evidence (p 15). Bruce Hay usefully distinguishes between the function of the judge and that of the jury (p 19). Peter Tillers chips in with a defence of the proper use of Bayes' Theorem (pp 20-21). David Kaye mentions what I have here called narrative evidence, at p 24. Ronald Allen comes down hard against Bayesianism (p 25) although he acknowledges it has some use (p 26). David Kaye brings narrative evidence into a Bayesian approach (p 29), and then Peter Tillers brings the discussion to cows (p 30) and common sense (the "stories" approach to probative value).

The reality is that juries are commonly told that they may decide to give particular evidence little or no weight (probative value) notwithstanding that it is (necessarily) relevant evidence. Evidence can be relevant although it has only a slight tendency to prove the matter contended for, and it will be admissible unless excluded by some other rule, and its probative value – assessed in the context of all the relevant evidence in the case – may be similarly slight yet its impact on the result of the case will depend on the priors (that is, how close the other evidence brings proof of the prosecution case to the required standard).

Bayesian reasoning can be useful on the issue of admissibility where it can be shown that the likelihood ratio is close to one (the evidence is nearly as consistent with innocence as it is with guilt) if other circumstances in the case make the evidence in question liable to exclusion because of its illegitimately prejudicial effect (see s 8 of the Evidence Act 2006; rule 403 of the Federal Rules). Evidence that is merely narrative should not have the necessary prejudicial effect to require exclusion, but it is commonplace to encounter exclusion of other relevant evidence because of its prejudicial effect.

Stories or statistics?

Usually people decide what to believe on the basis of what seems, without the need for further inquiry, to be consistent with common sense. They are using experience as the basis for judgment. They would have to concede that other people's experience can be useful in helping them make that judgment, and that that experience may come from statistical studies. The usefulness of scientific studies, the results of which are presented statistically, cannot be denied. They can distinguish factual from fictional stories. The significance of probabilities for logical reasoning must be recognised, and the inescapable influence of conditional probabilities on the correct determination of judicial proceedings must be utilised by fact-finders.

Accounting for guilty pleas – justice, not economics

Whereas South Australia v Totani (last note) illustrated the error of a legislature in depriving a court of its judicial function, Hessell v R [2010] NZSC 135 illustrates a departure by a court (here, the lower appellate court) from its judicial function.

This is only hinted at obliquely. The Court was highlighting the inappropriateness of a formulaic approach to the level of sentence discount to be awarded for entry of a plea of guilty. The Court of Appeal in this case had issued a guideline judgment on discounts for guilty pleas, setting out a sliding scale according to the stage of the proceedings at which the plea was entered. The Supreme Court overruled that:

"[67] The law reform agencies in the United Kingdom and New Zealand saw valid reasons to move to a more prescriptive and structured approach to giving credit for guilty pleas in sentencing. The Court of Appeal was persuaded by their reasoning. But in giving effect to their proposals, the Court of Appeal has underestimated the complexity of the issue including the potential of the changes to impact on the protected rights of persons charged with criminal offending. It is also inappropriate for a court to make changes in sentencing policy that would restrict the capacity of judges to determine sentences that are considered to fit all the circumstances of the case. Where the development of sentencing policy is motivated by a utilitarian calculus it may not be appropriate for judicial decision. Judges should show restraint in moving beyond the area mandated by existing legislation when exercising their sentencing powers. The ultimate difficulty we have with the Court of Appeal's approach is that it is not mandated by the Sentencing Act." [emphasis added]

The Supreme Court's approach recognises the need to avoid coerced guilty pleas, and to protect the defendant's right to dispute the facts and to challenge the admissibility of evidence.

Sentencing involves the consideration of a range of circumstances, one of which may be a guilty plea, and the significance of such a plea has to be assessed in all the relevant circumstances of the case. However, the Court added that discounts for guilty pleas should not exceed 25%, as remorse is a separate consideration for which more credit may be due.

The Australian approach (R v Wong [2001] HCA 64, (2001) 207 CLR 584 at [76] per Gaudron, Gummow and Hayne JJ; R v Markarian [2005] HCA 25, (2005) 228 CLR 357 at [37] per Gleeson CJ, Gummow, Hayne and Callinan JJ) was preferred to that in the United Kingdom, although New Zealand uses more of an "instinctive synthesis" of gravity and culpability than does Australia: Hessell at 55.

Fundamentally, the Court of Appeal had addressed the question from a point of view that was distorted by non-judicial concern with administrative efficiency.

When laws go crazy

People interested in whether a legislature may require a court to act unjustly may get some pointers from French CJ's judgment in South Australia v Totani [2010] HCA 39 (11 November 2010).

This case was decided in the context of Australia's constitutional arrangements, but some fundamentals of interest to everyone are evident.

French CJ began by referring to the assumption of judicial independence which underlies the constitution [1] and which has its origin in English law [59], where fairness and impartiality were recognised as characteristics of proceedings in courts of justice [60]. While the characteristics of a court cannot be defined in a single all-embracing statement, an essential is the conduct of an adversarial trial by an independent and impartial tribunal [62]:

"At the heart of judicial independence, although not exhaustive of the concept, is decisional independence from influences external to proceedings in the court, including, but not limited to, the influence of the executive government and its authorities [footnote: As to the multiple location of judicial decisional independence in separation-of-powers protections providing for "judicial independence" and within the rubric of "due process" and "the rule of law", see Gerangelos, The Separation of Powers and Legislative Interference in Judicial Process, (2009) at 8]. Decisional independence is a necessary condition of impartiality. Procedural fairness effected by impartiality and the natural justice hearing rule lies at the heart of the judicial process [footnote: For a recent discussion of the natural justice hearing rule in this context, see International Finance Trust Co Ltd [2009] HCA 49; (2009) 240 CLR 319 at 379-384 [139]- [150] per Heydon J.]. The open-court principle, which provides, among other things, a visible assurance of independence and impartiality, is also an "essential aspect" of the characteristics of all courts, including the courts of the States [footnote: Russell v Russell [1976] HCA 23; (1976) 134 CLR 495 at 520 per Gibbs J; see also at 505 per Barwick CJ, 532 per Stephen J; [1976] HCA 23]."

Important for this case, but also of potential general application, is the idea that a legal system cannot require a court to act in a way which impairs its "institutional integrity". In the context of Totani, this idea appears as [69(3)]:

"The institutional integrity of a court requires both the reality and appearance of independence and impartiality [Footnote: Forge [2006] HCA 44; (2006) 228 CLR 45 at 77 [66] per Gummow, Hayne and Crennan JJ, citing Ebner v Official Trustee in Bankruptcy [2000] HCA 63; (2000) 205 CLR 337 at 345 [7]- [8] per Gleeson CJ, McHugh, Gummow and Hayne JJ]."

And the fundamental nature of that requirement - the reality and the appearance of independence and impartiality - is historical, pre-dating Australia's federation [72]:

"...Forms of external control of courts "appropriate to the exercise of authority by public officials and administrators" are inconsistent with that requirement [Footnote: Gypsy Jokers Motorcycle Club Inc v Commissioner of Police [2008] HCA 4; (2008) 234 CLR 532 at 553 [10] per Gummow, Hayne, Heydon and Kiefel JJ ]. The requirement is not a judicially generated imposition. It derives from historically based assumptions about courts which were extant at the time of Federation."

This form of argument is potentially a powerful weapon against those who would support the idea that a sovereign legislature may enact any law it wishes. I have a written a brief note on this sort of thing in "Criminal Equity" [2000] New Zealand Law Journal 427. By "criminal equity" I mean the pursuit of fairness in the criminal context by looking beyond the black-letter law. Since then, we have enacted a potential cause of unfairness in s 21 of the Evidence Act 2006[NZ] which is absolute in its terms. It is wide enough to prevent admission of the exculpatory parts of a defendant's statement to the police unless he gives evidence at his trial. In cases where this would operate unfairly, the court has to, in effect, disregard s 21, but, since the court does not have the power to rule the legislation unconstitutional, this is done under the guise of recognising an implicit power to require the prosecution to lead the exculpatory statement to prevent unfairness. In a case that seems to be currently subject to a suppression order, [2009] NZCA 607, (2009) 24 CRNZ 527, the Court of Appeal has said, at [19]:

" ... such a power must be implicit in s 25 of the New Zealand Bill of Rights Act 1990. In the event of an unfair decision by a prosecutor not to lead evidence as to what an appellant said at a police interview, it would thus be open to the Court to require that evidence to be led. As to this, see Illingworth and Mathias “The Admissibility of Hearsay Statements and Opinion Evidence” in Young and Chambers Evidence Act 2006 (NZLS Intensive June 2007) at 50 – 51."

Essentially this is using the inherent power to prevent an abuse of process to qualify the statutory language in the interests of trial fairness. The jurisdiction to prevent an abuse of process is inherent in the function of the court.

Fair multiplicity

A routine point of criminal law was the reason for the unopposed allowing of the appeal in Mason v R [2010] NZSC 129 (3 November 2010): s 329(6) of the Crimes Act 1961 provides that "Every count shall in general apply only to a single transaction." Here, the transaction covered two assaults, and these should have been charged separately to avoid unfairness: the accused may have had a defence to one assault, and on sentencing it would be necessary to know what had been proved beyond reasonable doubt especially because the assaults varied in seriousness.

The conviction was quashed and the Crown did not seek a retrial.

This rather elementary point had been overlooked in the Court of Appeal, where the appellant had represented himself and the Court had misunderstood his argument as applying only to the determination of facts for sentencing.

Mr Mason will no doubt be pleased he did not have to go to the trouble and expense of trying to appeal to the Privy Council. This was a minor case, alleging punching in the face and ear-pulling, committed against one of the accused's sons. It was only interesting because of the huge attention that the use of force in domestic discipline has recently received, and this was regarded as a test case for new legislation, s 59 Crimes Act 1961.

There are occasions where a count can allege offending over an extended period, and there may be many alleged incidents which a complainant cannot be specific about. In those cases (often, but not necessarily, alleging sexual misconduct) the jury must, before it can find guilt, agree on at least one incidence of the alleged offence.

A leading authority is R v Accused (CA160/92) [1993] 1 NZLR 385; (1992) 10 CRNZ 152 (CA), where the judgment of the Court of Appeal was delivered by Cooke P (later, Lord Cooke of Thorndon). On the requirement of trial fairness, his Honour said

"The basic ingredients of a fair trial remain. There are limits to custodial interrogation and in some circumstances Bill of Rights protections; the accused is entitled to know the substance of that with which he is charged; he has statutory and common law rights regarding the disclosure of certain information; he has detailed notice before the trial of the evidence to be called by the prosecution; he has the right at the trial of cross-examining the Crown witnesses, subject to some reasonable restraints; he has the right to give evidence himself, so that the jury may compare his evidence with that of the complainant or his other accusers; and he has the benefit of the doubt, invariably underlined by trial Judges in emphatic directions that the prosecution must establish its case beyond reasonable doubt. In the event of a conviction the accused has the right of appeal on grounds including the basic one, miscarriage of justice. There is the safety net for exceptional cases of the Royal prerogative and s 406 Crimes Act: see Burt v Governor-General unreported, 16 July 1992, CA175/88.

"On the general question of the fairness of criminal trials in New Zealand it is not to be overlooked that other developments, some of them with no particular bearing on sexual charges, have also moved the balance towards the prosecution. These include statutory provisions for electronic surveillance; DNA testing; a more liberal judicial attitude to "similar fact" evidence and hearsay evidence. But they have been accompanied, at least since Police v Hall [1976] 2 NZLR 678  and R v Hartley [1978] 2 NZLR 199, with affirmation of the Court's inherent jurisdiction to prevent unfair trials; and that jurisdiction would be available if truly needed in a case in the present field.

"It is possible to imagine a case in which allegations of sexual misconduct are so vague or relate to a time so long ago, without justification for the delay, that it would be unfair to place an accused on trial upon them. Then the possibility of exercising the protective inherent jurisdiction would fall for consideration in all the circumstances of the particular case."

This comes down to the avoidance of vagueness and the avoidance of prejudicial or unconscionable delay. Logical difficulties may persist, as mentioned in the discussion of the use of specimen or sample counts in Tyack v. The State (Mauritius) [2006] UKPC 18 (29 March 2006) at 18-21. The problem is one of framing legislation that permits inclusion of more than one occurrence of an offence in one count, while still complying with the right to a fair trial. The New Zealand provision, s 329(6) above, is not absolute, but applies "in general" to a single "transaction". In Mason the Supreme Court said of this:

" [9] ... The qualification "in general" and the relatively indefinite word "transaction", which can encompass both a single event or a course of conduct, recognise the difficulty of application of any precise rule to the charging of the many different fact situations in which acts of offending may occur. They indicate the need for some flexibility. The essential requirement emerging from case law is that, if particular acts of alleged offending can sensibly be charged separately without undesirably lengthening the indictment (overcharging), then that should be done. It is necessary that distinctly identifiable acts of alleged offending be the subject of separate charges where the accused may be prejudiced either at trial or on sentencing if they are combined in a single count. On the one hand, the use of a multiplicity of counts is to be avoided where fewer would suffice for the interests of justice. On the other, overly complex counts may prejudice the defence or make it difficult to frame fair and accurate directions to the jury. If necessary trial Judges should intervene if either problem arises.

"[10] We repeat what Anderson J said for this Court in R v Qiu [[2007] NZSC 51, [2008] 1 NZLR 1 at [8] ].The Court endorsed the practice of not charging as separate offences a continuing course of conduct which it would be artificial to characterise as separate offences. But the Court said that it was another thing to charge as a single count repetitive acts which can be distinguished from each other in a meaningful way, even if they relate to more than one act of a certain class or character. The Court added something which the present case vividly illustrates:

'Separate counts facilitate fairness in the conduct of the trial by focusing attention on matters of fact and law which can and need to be distinguished for the purposes of different counts. In the event of conviction, they assist the sentencing Judge by indicating the extent of culpability.'"

Some delicate calculations may need to be made, as appears to have been the position in Mason, where experienced defence counsel had not sought severance of the count. Presumably this was because the chances of a successful defence to the lesser allegation were thought to be low, and that even if successful, splitting off the lesser allegation would merely highlight the more serious one and would prevent a sentencing submission that the facts should be assessed as less serious than they could have been. Fairness to the defence is not always easy for the defence to accept, as was seen earlier in these notes in relation to leaving alternative counts to the jury: "Fairer than you may want" .

Good character direction

An addition to our collection of cases illustrating when absence of a good character direction requires the quashing of a conviction: Campbell v R [2010] UKPC 26 (3 November 2010).

Here the omission of the direction was not the result of a tactical decision by the defence, and the case involved a contest between an eyewitness and the accused who gave evidence. In such cases the direction would be "of greatest potential significance" (para 45).

The Board also summarised the requirements for an eyewitness identification direction (para 28), and began with an extended discussion of when the Board can grant special leave to appeal to it where the lower court has refused leave (para 5-25).

W(h)ither Weiss?

Usually only one side in a criminal case appeals, but in R v Nguyen [2010] HCA 38 (3 November 2010) the Crown appealed against the Court of Appeal of the Supreme Court of Victoria's quashing of convictions for murder and attempted murder and its entry of acquittals, and the accused (DQN) appealed on the basis of an alleged misdirection by the trial judge by way of failure to leave to the jury the alternative of manslaughter on each count. DQN's appeal point had been raised in the Court of Appeal but was not addressed in its judgment.

The High Court therefore had to consider, on the Crown's appeal, whether the Court of Appeal had been correct in holding that the verdicts had been unreasonable, and on DQN's appeal, whether there had been an error of law.

Both these appeals had to be decided: it would have been insufficient to simply dismiss the Crown's appeal because there had been a substantial miscarriage of justice, because that would leave standing the Court of Appeal's entry of acquittal.

In the result, the Crown was entitled to a retrial because the verdicts were not unreasonable and the Court of Appeal should not have allowed DQN's appeal on that ground. There had to be a retrial because DQN's appeal also had to be allowed: there had been a substantial miscarriage of justice at trial.

Do these two grounds of appeal – reasonableness of verdict, and error of law giving rise to a miscarriage of justice – require different approaches by the appellate court? Both are subject to the proviso that the appeal may be dismissed if no substantial miscarriage of justice had actually occurred.

What is interesting about the High Court's judgment is its absence of reference to Weiss v R (2005) 224 CLR 300. Weiss requires the appellate court, upon finding that there was an error at trial giving rise to a miscarriage of justice, to determine whether the miscarriage was "substantial" by examining the whole of the record to assess whether the prosecution case was proved beyond reasonable doubt. Should you wish, you may locate my earlier comments on this by searching this site on the word Weiss.

In Nguyen the Court applied M v R [1994] HCA 63, (1994)181 CLR 487. This involves asking "was it open to the jury" to convict? It differs from the Weiss question, "is the appellate court satisfied of guilt"?

Plainly, the High Court could not apply Weiss because that would involve saying that it was satisfied of guilt, and also that there had been a substantial miscarriage of justice arising from failure to leave the jury with the alternative of manslaughter. The implications of this appeal highlight how inappropriate the Weiss approach can be.

Classic probability conundrums

There are many examples online of probability conundrums. I have mentioned some in my draft paper on propensity evidence, linked on this page. I mention two of them here.

The first problem is, if you are told that a stranger to you has two children, at least one of whom is a boy, what is the probability that the other is also a boy?

Note that neither child has been singled out, so this is a problem about the probability of having two boys. Given that one child will have been born before the other (even if a twin), the two children can have arrived as BB, BG, GB. The other combination, GG does not count in this example. So, one out of three eligible combinations gives two boys, and the probability of that occurring is 0.33.

If one child had been singled out, the problem would have been about the gender of one child (the other one). You might have been able to see one child, and were asked about the other. For one child, the probability of it being a boy is 0.50.

This problem illustrates how important it is to ascertain exactly what the issue is. The seemingly endless argument about this problem in the discussion to the above NY Times article exemplifies this.

Some people wonder why the combinations BG and GB are counted separately. They would say that the relevant combinations are just two: BB and (B and G). The error here is in thinking (B and G) will occur just as frequently as BB. This overlooks the way in which the data can arise. (B and G) will occur in two ways (namely, BG and GB) whereas BB will occur in only one way.

So the two lessons from this problem are: ascertain the issue, and examine how the data arises.

The other problem is known most commonly as the Monty Hall problem, after a game show host. The task is to pick which of three doors, A, B and C, will when opened reveal a prize. You pick one door, say A. The host, who knows where the prize is, then tells you it is not behind door C. Should you change your guess to door B?

When you choose door A, you divide the doors into two groups or classes: the chosen and the not-chosen. The probability of the prize being behind door A is 0.33. The probability of it being in the not-chosen class is 0.67. Once the host eliminates one member of the not-chosen class, the class probability for that class attaches to the only remaining member, door B. You should change your guess.

Some people object that each door always has the same probability of concealing the prize, and that there is no reason to change your guess from A. This ignores the new information the host gives you. Changing the probability distribution among members of one class does not affect the probability distribution among member(s) of another class. The constant probabilities are wrongly linked to the individual doors, rather than to the classes.

The lessons from this example are: use all the relevant information, and recognize when the issue is about members of one class as distinct from members of another class.