It just struck me that I assumed every card has a letter on one side and a number on the other, which was not specified.
I adjust my answer. You need to turn A, K, and 7.
Edit: And I see I’m on the heels of @Riemann !
I got AK7 immediately. I’ve seen similar puzzles where it was specified that there was a letter on one side and number on another, so I re-read the description carefully. Only when we see an even number can we be certain that the other side doesn’t matter.
Maybe the poll should have had 16 entries…
The rule can be rewritten as “Odd numbers never have a vowel on the other side”.
But turning over the AK (assuming face cards are all not even?) and 7 only proves the rule for the universe of those four cards. If the rest of the deck is laid out next to those four, you haven’t proved the rule for any of those cards (which is just an example of not being able to prove a negative).
Interesting. The official right answer apparently neglects that vowel vowel possibility.
I’ll link later but not before giving more examples from the article! 
You are assuming that there is always a letter on one side and a vowel on the other, that’s not stated. It’s not even stated that there’s only one thing on each side side of a card. When you turn over the K you might see the entire alphabet and every number from 1 to 9 all there.
If you want to rewrite the rule, it’s:
“When an even number is absent, there is not a vowel on the other side.”
The rule doesn’t say anything about the existence of a deck, let alone any rule that might apply to it. It is stated as a rule for these 4 cards.
It certainly doesn’t say that there’s always a letter on one side and a number on the other, so if you turn over K and find a vowel, the rule is broken.
In fact, it doesn’t place any limits at all on what could be on any one side of a card. The other side of the K card could have the whole alphabet and every number from 1 to 9 plus a picture of a cute octopus on it. On a plain English interpretation, that would satisfy the “if there is a vowel” condition and break the rule.
Similarly, a plain English reading of the question does not imply that the presence of a vowel implies that an even number is the only thing on the other side. “There is an even number” just says that an even number must be present, perhaps along with other things.
It’s like the “at least one side of at least one sheep in Scotland is black” joke.
Another of the classic list:
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Which statement is more probable?
(a) Linda is a bank teller (B).
(b) Linda is a bank teller (B) and is active in the feminist movement (F).
- A
- B
0 voters
You bunch are like the one who answers the lily pad question with “one day”.
They didn’t say which half. The last half is covered in one day.
I suspect most of this group will get the list I’m going through right. What’s interesting to me though is not that so many of the general public in the studies do not, but why they do not.
The other two:
The probability of breast cancer is 1% for a woman of a particular age group who participates in a routine screening. If a woman who participates in a routine screening has breast cancer, the probability is 80% that she will have a positive mammogram. If a woman who participates in a routine screening does not have breast cancer, the probability is 10% that she will have a false-positive mammogram.
What is the probability that a woman of this age group who participates in a routine screening and has a positive mammogram actually has breast cancer?
And
A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50 percent of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50 percent, sometimes lower.
For a period of 1 year, each hospital recorded the days on which more than 60 percent of the babies born were boys. Which hospital do you think recorded more such days?
The larger hospital
The smaller hospital
About the same (that is, within 5 percent of each other)
(The article also includes The Monty Hall Problem but we can skip it.)
The article explores how various factors impact performance.
The screening one is nearest and dearest to my heart. It is surprising how many of those who should know better make the common errors in my real world.
Agreed, to take it back to Tim Harford and the original video, regardless of whether people think he did a good job of presenting that problem, his day job is all about highlighting similar statistical issues in the news and he does it really well.
A typical problem is the relative v absolute risk.
Headlines might state “eating x doubles cancer risk” when what is actually shown is very different.
The study actually may be reporting a 25-45% increase in lifetime risk for a specific cancer. That cancer may have a lifetime risk of 1 in 500, the behaviour under study may raise that lifetime risk to something like 1.25 or 1.45 per 500 people.
Giving people the awareness and the tools to dig a little deeper helps raise understanding and, one would hope, make people a little less likely to be taken in.
I like this one, it is a really good example of where re-formulating your answer in different ways and stating it out loud can help flush out an error.
If say (B) what am I actually saying? I’m saying that there are more universes in which she is both things than there are where she is only a teller. But seeing as every universe we are talking about here must already have Linda as a bank teller, how can (B) make sense?
Or perhaps the benefit of restating the problem in simpler terms.
A six sided dice has 5 faces with a green 2 on it, and one face with a red 4. We roll it, what is more likely?
A) That it shows an even number?
B) That it shows an even number and that number is green?
Stated in that way it is far more obvious that (A) is more likely. Obviously there are no instances in which the condition of option (A) is not satisfied but the same cannot be said for the second condition of option (B)
What really interests me is how best to equip people with the tools in order to interrogate these problems and situations. How best to increase the probability that people will get these problems right.
And this relates the article I linked to, to the video of the OP.
The article begins with a great review of the classic sets of cognitive illusions, and some discussion about thoughts of their bases, but it goes on with an experiment using students 16 to 18 years old in Luxembourg.
Higher math literacy helps. Higher reading literacy helps. Overall higher intelligence helps but less. And presenting the essence of each problem in a manner that makes them more accessible and less confusing - changing the representation of the presented information - such as changing 80% to 8 out of 10. Those with lower math and reading literacy were helped the most by the clearer presentation, and clearer presentation helped those with higher literacies the least.
Relevant to the video, a less clear presentation made even those with higher math and reading literacy subject to the illusions.
Personally I am a believer in critical thinking “habits of mind” and I believe that those “habits” involve more than math and reading literacy and do not necessarily correlate with intelligence overall. I expect that many attracted to this site, and this thread, have those habits of mind. I am not sure how they best get taught to, or encouraged in, a general population. Yes learning specifically and explicitly about probability and Bayesian reasoning is important. Math and reading literacy in general too. “Cognitive reflection” as presented, as a solution for developing those critical thinking skills, seems however to be a bit simplistic, as does any broad brush dismissal of heuristics in general.
BTW did you answer the breast cancer screening one? Looking out for that illusion is important in many venues, not only medicine.
I agree with all your points, there are lots of potential approaches and I do bemoan the fact those “habits of mind” you mention were not (and still are not) taught enough in UK schools. My wife and I try to instill it in our own kids, she has a mathematics background but others sadly miss out.
I did but I struggled (probability calculations are not my strong point) and so asked my wife for help.
I did get the first part straight in my head, i.e. that it is important to seperate out the number in a population who have cancer and who don’t. So for a population of a thousand we have 10 who have cancer and 990 who don’t.
I was about to head in roughly the right direction but at that point I got my wife to check it with me and give me pointers.
So the next step is, of the 10 who do have cancer 8 will get a true positive and 2 will be a false negative.
Of the 990 who don’t have cancer, we would see 99 get a false positive result.
So in total, we have 107 positives, only 8 of those are for women who do have cancer which is about 7.5%
Thanks to Mrs Novelty Bobble who has the patience of a saint and introduced me to a form of tree diagram that helps to set out such problems.
Seeing it sketched out certainly helps, as does inserting other values into the calculation, seeing what happens and asking if the conclusion I came to still makes sense.
i.e. if I made the false positive rate absolutely tiny what would happen? the probability of actually having cancer given a postitive test would be very high, that checks out.
The tree and the conversion into real people numbers (e.g. out of a thousand) are part of the methods used in the study as ways to present in a more understandable manner.
A key takeaway is the even tests with very good sensitivity and specificity may result in more false positives than true positives if the a priori for the condition is infrequent enough.
The Bayes method btw is not dissimilar to what we used to spell out the bit about reflection being associated with a greater chance of being incorrect.
This is where the public understanding of science and probability more generally could do with a boost.
Take the calculation regarding the screening test we just did and imagine the sensationalist headline
“Cancer test is only 7% accurate”
It is defensibly true from a certain perspective but in essence it is horribly wrong.
I’m not sure many publications or news outlets would actually take the time to describe accurately what was really going on and I’m not convinced the general public are knowledgeable enough to see the linguistic sleight of hand.
No, it is not horribly wrong at all. “Sleight of hand” would be to look only at the characteristics of the test and to ignore the incidence of the disease. It is potentially a real problem that the false positive rate here is ten times the incidence of the disease. If the standard of care after a positive test would be (say) a biopsy that entails some risk, pain and financial cost, then we have to weigh this penalty for the 92.5% who get a biopsy they don’t need vs the benefit for the 7.5% who get an early cancer diagnosis. This is how to decide whether routine screening is beneficial.
The headline is horribly wrong because a woman could easily take from it that the test will detect their cancer 7% of the time when in reality it is far, far more accurate than that.
I still feel that you don’t get it at all. The question of how reliably a test will detect cancer is the wrong metric when deciding whether you should get screened.
What if only one person in the history of mankind had ever had a disease? Should we still screen for it, just because the test is very good at detecting this disease, when it is present? Suppose you test positive, will you get a painful biopsy to rule it out, even though after testing positive the probability you have the disease is STILL only one in a billion?
Yes if the headline confused that with sensitivity that would be unfortunate.
OTOH @Riemann’s point is not only important, it also harkens back to the issues of the video!
The balance that would go into that calculation requires placing some value on potentially missed cases and some presumptive increased number of deaths, against a larger number of individuals having real harms and costs from biopsies. How many of one “major” are offset by how many of another “minor”?
There is a belief commonly held that more testing is good, even if the odds of the condition being tested for is low. Understanding how Bayes applies explains why it may not be better to get a test without good reason: the ratio of false to true positive may be very very high and the process of determining that can cause real harms and costs that in aggregate outweigh potential benefits.
I disagree, if there are downsides to taking the test and risking the implications of a false positive then one key thing to weigh in the balance is how accurate the test is. To deliberately misrepresent that part of the equation, to misinform, to make it the most sensational and memorable takeaway for people seems to me to be ethically wrong at best.
The headline as given is grossly misleading as to the ability of the screening process to detect cancer. It should avoid doing that.
Are you suggesting that headline, as written, is the headline message you’d want to people to take away? You want them walking away thinking
“shit, even if I do have cancer there is only a 7.5% chance that the screening will pick it up, that test is utter crap”
For me, the headline should not deliberately misrepresent the screening test. Maybe that’s just the pedant in me.
Should an article also go into the risk/benefit analysis of the screening process and implications of disease rates v false positives and negatives? In my ideal world yes. If the article wants to start with a headline of that nature and go into the level of detail we have here then even better. Just don’t start with a willfully misleading headline.