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June 24th, 2018 (Permalink)

The Four Ace Puzzle

He's back: "Three Card" Monty, the carnival mountebank who does not manipulate cards―he manipulates minds.

"Attention, Suckers!" Monty called out from his booth to the passing crowd on the boardwalk. "Um, I mean, attention, ladies and gentlemen!"

One hapless fellow slowed his steps, glancing Monty's way. "You, sir! Are you a gamblin' man or just a ramblin' man?"

"Um, er," the man replied.

"Excellent!" Monty continued, "What is your name, my good man?"


"Well, step right up, John, and try your luck." John walked hesitantly toward the booth. Monty picked up a deck of playing cards and dealt out four from the top of the pack face up in a row on the counter in front of John. The four cards happened to be the aces.

"Now, you see before you four cards: the Ace of Hearts, the Ace of Spades, the Ace of Clubs, and the Ace of Diamonds. Two of the cards are red―the Ace of Diamonds and the Ace of Hearts―and two are black―the Ace of Clubs and the Ace of Spades.

"I turn the cards over, gather them together, and shuffle them", Monty said and did so. "I deal them out in a row again, but this time face down", he said and did.

"In a moment, I will let you choose two of these four aces. My question for you is this, John: What is the probability that the two cards you choose will be the same color?"

"I'm not sure," John replied after a pause.

"They're either the same color or they're not, right?"

"Of course."

"And there are two cards of each color, so it's fifty-fifty odds. That means you have an equal chance of choosing cards of the same or different colors. Don't you agree?"

"That sounds right."

"So, if I were to offer you an even money bet that the two cards you select will be different colors, that would be a fair bet, wouldn't it? In other words, I'll bet you dollar for dollar that the two cards you choose will differ in color. If they're the same color you win, but if they're different colors I win. That's fair, ain't it?"

"I guess."

"So, John, how much do you wish to wager?"

Should John accept Monty's wager, that is, is the bet indeed a fair one?


June 22nd, 2018 (Permalink)

Fact Vs. Opinion

The Pew Research Center performed a large survey earlier this year to find out whether Americans can tell the difference between factual statements and opinion statements1. Included in Pew's reporting of the results of the survey is a short quiz where you can test your own ability to use this distinction. I suggest taking this self-test before you continue reading2, as I discuss the distinction below and don't want to influence your results.

Welcome back! I hope you did well, but then I expect that readers of The Fallacy Files are better educated on this subject than the average person. If your results were disappointing, all the more reason to keep reading and learn this important distinction.

What is a fact? In this era of fake news and fact checking, it's valuable to be able to tell the difference between factual reporting and expressions of opinion. One common form that propaganda takes is opinion masquerading as straight reporting, so an important skill for unmasking propaganda is to be able to tell the difference.

Unfortunately, the Pew poll seems to indicate that people do only slightly better than chance in telling the difference between factual statements and opinion. The results show that greater education is correlated with greater ability to tell the difference between factual statements and opinion3, but either education doesn't make a big enough difference or it's not reaching enough people.

One thing that may have affected the survey's results is confusion about the difference between a fact and a factual statement. Some people may well assume that these are the same thing, but they're not, at least not for the purposes of this survey.

First of all, what is a statement? It's a sentence that is true or false. Not all sentences are true or false; for example, neither questions nor orders are statements. A fact, is at the very least, a true statement but there's more to it than that4, which I'll discuss below. A factual statement is a statement that, if true, would be a fact. So, a fact is a true factual statement.

This means that there can be false factual statements, which will sound like an oxymoron if you think that fact = factual statement. Anyone thinking this while taking the Pew survey or quiz could be led to misclassify false factual statements as opinions6. The survey and the report on it are actually fairly good at explaining this; for instance, the report defines the difference between these two types of statement as: "…factual―something thatís capable of being proved or disproved by objective evidence―or…an opinion that reflects the beliefs and values of whoever expressed it.5"

Factual statements, not facts, are actually what fact-checkers check. There is no need to check facts, since they are true. What we need to check is a factual statement, which may be true or false, in order to find out which it is.

What, then, is a statement of opinion? Clearly, it is a type of statement, which means that opinions are either true or false. So, a question or an order is not an opinion. The difference between matters of fact and opinion is not black or white, but a scale with fact at one end, opinion at the other, and a gray area in between. Roughly speaking, a factual statement is a statement whose truth-value any competent researcher ought to be able to determine. In other words, "factual statement", in this sense, is the sort of statement that a fact-checker would be able to check. For instance, one of the statements in the quiz is: "Spending on Social Security, Medicare, and Medicaid make up the largest portion of the U.S. federal budget." This is a factual statement because anyone ought to be able to check the federal budget to see whether it's true, and it's a fact because it is true.

However, not all factual statements are true. For instance, the negation of the above quiz question is also a factual statement: "Spending on Social Security, Medicare, and Medicaid do not make up the largest portion of the U.S. federal budget." This is a false factual statement that can be checked in exactly the same way as the affirmative statement.

In contrast, consider another statement from the quiz: "Democracy is the greatest form of government." This is a statement of opinion, rather than a factual statement, because it involves a value judgment. You may think it's true, and it may even be true, but it's not a fact because it's not a factual statement. Statements of value, in general, are matters of opinion rather than of fact.

As an exercise, the next time you read a news report, try classifying every statement in it as either a factual statement or a statement of opinion. Has the reporter kept to factual claims, or have matters of opinion been included among the facts? If the latter, do the opinions expressed support one side of a controversial issue?

Reader Response (6/29/2018): Lawrence Mayes emails:

Pew classify "ISIS lost a significant portion of its territory in Iraq and Syria in 2017" as a factual statement. I disagree. If it had said: "ISIS lost 10% by area of its territory in Iraq and Syria in 2017" I would agree that it's a factual statement. Perhaps in the US the phrase "a significant portion" has a very precise meaning7; however, I cannot see that the phrase can be anything other than a matter of opinion. And it is a matter of opinion whether 10% (from the above example) is significant and, if not, what area could be regarded as significant. A 10% loss may not sound significant but it could be argued that it is. It could be said that it certainly was significant as the 10% area comprised 90% of the population and all the oil wells (or satisfied any other of an indefinitely large number of distinct definitions of "significant portion" in this context). But a counter-argument is that only a loss of more than, say, 40% is significant. Another that only 1% is. And so on.

Good point! I agree that what is "significant" is usually a matter of opinion for three reasons:

  1. It's vague, for the reasons you've indicated. Pew, in its discussion of the distinction, mentions vagueness as one difference between factual statements and statements of opinion.8
  2. "Significant" represents a value judgment that the area of territory lost, or perhaps its population or economic value, is of importance.
  3. Despite the above points, however, I think it is a fact that ISIS has lost a significant portion of its territory in the last few years: according to one source9, it has lost 98% of the territory it held at its peak, and a comparable percentage of members. Despite the vagueness of "significance", this has got to be a "significant" loss.

    Just because a statement contains a vague word does not mean it cannot be factual. For instance, consider "baldness", the paradigm example of a vague term. It's a fact that Telly Savalas was bald. It's only when dealing with borderline cases that it becomes a matter of opinion. So, a few years ago it would have been a matter of opinion whether ISIS had lost a significant amount of territory, but it is no longer.

    However, to realize this one would have to know that ISIS has lost almost all of its territory, and Pew did not intend for its survey to be a test of factual knowledge; rather, it was supposed to test one's understanding of the distinction between factual and opinion statements2. It's possible that some survey-takers who lacked this knowledge classified the statement as a matter of opinion, which may have contributed to the poor results. So, for this reason I agree with you that the statement should not have been included in the survey.


  1. Amy Mitchell, Jeffrey Gottfried, Michael Barthell & Nami Sumida, "Distinguishing Between Factual and Opinion Statements in the News", Pew Research Center, 6/2018 (PDF).
  2. "Quiz: How well can you tell factual from opinion statements?", Pew Research Center, accessed: 6/20/2018.
  3. Amy Mitchell, Jeffrey Gottfried, Michael Barthell & Nami Sumida, "2. The ability to classify statements as factual or opinion varies widely based on political awareness, digital savviness and trust in news media", Pew Research Center, 6/18/2018.
  4. "Fact", like most words, is highly ambiguous, and some philosophers and logicians do treat it as a synonym for "true statement". See, for example, Monroe C. Beardsley, Thinking Straight: A Guide for Readers & Writers (1950), p. 5.
  5. Amy Mitchell, Jeffrey Gottfried, Michael Barthell & Nami Sumida, "Distinguishing Between Factual and Opinion Statements in the News", Pew Research Center, 6/18/2018.
  6. Clarification (Added: 6/23/2018): This couldn't have happened in the present survey, since Pew chose to include only true factual statements, for some reason. However, anyone who mistakenly thought that one of these statements was false might be led to conclude that it must, therefore, be an opinion rather than a fact. By choosing to include only accurate factual statements, Pew may have caused this source of confusion to be underestimated by the poll results. I'd like to see a version of this survey in which half of the factual statements were true and half false to see what effect on the overall results this change would have, as well as whether there would be a difference between how the two types of factual statement were evaluated. See: Amy Mitchell, Jeffrey Gottfried, Michael Barthell & Nami Sumida, "Methodology", Pew Research Center, 6/18/2018.
  7. No, it doesn't.
  8. Amy Mitchell, Jeffrey Gottfried, Michael Barthell & Nami Sumida, "1. Overall, Americans identified more statements correctly than incorrectly, but sizable portions got most wrong", Pew Research Center, 6/18/2018.
  9. See, for instance: Jamie McIntyre, "Here's how much ground ISIS has lost since Trump took over", Washington Examiner, 12/23/2017.

Solution to the Four Ace Puzzle: John should definitely decline this sucker bet. The odds are actually two to one in favor of Monty, that is, the probability of selecting at random two cards of the same color is only one-third. So, an even bet is not a fair bet; instead, for a fair bet, Monty should put up two dollars for every dollar bet by John. If John accepted Monty's wager, then in the long run Monty would win two plays of the game for every one won by John. So, Monty would win two dollars for every dollar he lost, and John would lose two for every one he won.

An easy way to see that this is the case is to list every pair of cards that can be made from the four Aces. There are only six such pairs, disregarding the order in which the cards were selected, which doesn't affect whether the colors are the same or different:

  1. Spade, Club: Same color
  2. Spade, Heart: Different color
  3. Spade, Diamond: Different color
  4. Club, Heart: Different color
  5. Club, Diamond: Different color
  6. Heart, Diamond: Same color

As you can see, two of the six pairs are the same color and the remaining four are different colors. So, if you choose two cards at random, the probability of getting a pair that is the same color is only one-third, which means the odds are one to two, not even.

As a general rule, you should never make a bet with Monty unless you want to lose.

Note: I came across a version of this puzzle in Ian Stewart's Professor Stewart's Hoard of Mathematical Treasures (2009), pp. 159 & 316. This puzzle is a work of fiction. Any resemblance to actual events, locations or persons, living or dead, is entirely coincidental.

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