Tuesday's child
The editor writes:
鈥� Alex Bellos reported a probability conundrum posed by Gary Foshee at the recent Gathering for Gardner event: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?” (29 May, p 44). We have been deluged with contributions about it, with almost everyone arguing that our solution of 13/27 is incorrect. Tet Woo Lee has provided a clear explanation of the solution.
From Robin Russel
Bellos states that the possible combinations of two children are GG, GB, BG and BB. This implies that the children’s birth order is distinguishable by gender only, and not by the fact that one child has been labelled (for example, by the day of the week he or she was born on). However, Bellos later lays out a list of 28 potential combinations which explicitly distinguish between the labelled boy being born before or after his male sibling.
Either the possible outcomes are distinguishable by gender and not label 鈥� giving 21 possible outcomes (7xGB, 7xBG and 7xBB) and producing a final answer of 1/3, or they are distinguishable both by gender and label 鈥� giving 28 possible outcomes and an answer of 1/2. Birth order is either label-distinguishable or label-indistinguishable: either way it must be consistent in both parts of the reasoning for the puzzle.
Walton-on-Thames, Surrey, UK
From Ron Wells
The overall human sex ratio at birth averages about 51 per cent in favour of male children, but is highest for first-born children and less so for each successive child. There is also a slight tendency for successive children to be of the same sex. So, if one of Gary Foshee’s two children is a boy, and in the absence of other information on factors in the Foshee family known to affect the birth sex ratio, the other Foshee child whether born before or after that boy has slightly more than a 51 per cent chance of also being a boy. Of course the days of the week on which births occur has nothing to do with it.
Mawson, ACT, Australia
From Tet Woo Lee
Among all parents who have two children, one quarter of them will have two boys, and the probability of a second boy being born if you already have a boy is 1/2. However, if you state that at least one of two existing children is a boy, the probability of the other being a boy becomes 1/3, because you have to exclude all those who have two girls from the population of parents in question. You are not changing the probability of having a boy in future, nor are you changing the probability of having two boys if you have two children. What you are changing is the group of parents we are talking about.
The probability of having two boys when you have at least one boy born on a Tuesday approaches 1/2. You have enriched the parent population by excluding all those who do not have at least one boy born on Tuesday from the population of interest 鈥� those with two girls, those with one boy but not born on Tuesday, as well as those with two boys but neither born on Tuesday. While there aren’t as many parents with a boy born on a Tuesday as those parents with any two children, a greater proportion of parents with a boy born on a Tuesday have two boys compared to parents with any two children.
As stated in the article, if you include a rarer piece of information, you enrich the population further and get even closer to 1/2. You cannot exceed 1/2 because there are always going to be at least as many parents who have one boy with a particular criterion plus a girl as those who have two boys, one of which meets that criterion.
While the probabilities calculations in the article are correct, one valid criticism remains: what is the point of asking such a question?
Henderson, New Zealand
Vivisection value
Simon Festing says that reducing publication bias in animal research would ensure a sound basis to move from animal studies into clinical trials (5 June, p 22).
This would be true if the results of animal studies translate directly to humans. They do not, which is a far more important problem than publication bias.
Full publication of every animal study of the immunomodulatory drug TGN1412, for example, would still have suggested that it was safe to proceed to clinical trials, since the devastating response to the drug is unique to humans.
Systematic reviews of the applicability of animal results to human medicine 鈥� such as those by Pablo Perel and others () and by Daniel Hackam and Donald Redelmeier () show consistently that animal studies predict human response incorrectly a majority of the time. In the case of stroke, is anyone seriously suggesting that more than 150 treatments successful in animals have failed in humans because of publication bias?
Of course, this bias should be addressed. There should unquestionably be a registration system for animal studies, as there is for human studies. As chief executive of Understanding Animal Research, surely Festing should be calling for this, rather than merely commenting that “it is not inconceivable that we might move towards a similar system”.
From Marshall Deutsch
In her article “Eat less, live longer?”, Laura Cassiday points out that followers of a restricted diet mostly “hover around the lower limit of ‘normal’ body mass index [BMI], at 18.5 kilograms per height-in-metres squared” and have low cholesterol levels (29 May, p 36).
However, many studies, such as that by Katherine Flegal and colleagues in the The Journal of the American Medical Association (), show that mortality associated with weight is at a minimum in people with a BMI of 25 to 29.9, a range considered overweight.
Similarly, many studies show that older people with high blood cholesterol live longer than those with low blood cholesterol. One example is the upcoming work by P盲ivi Tuikkala, of the University of Eastern Finland in Kuopio, and colleagues, who conclude: “Participants with low serum total cholesterol seem to have a lower survival rate than participants with an elevated cholesterol level, irrespective of concomitant diseases or health status” ().
But don’t animal studies show that dietary restriction leads to a longer life? Yes, if those little furry prisoners are denied exercise privileges. Jean Mayer, in his book Overweight: Causes, cost, and control, showed rats that exercised from 1 to 8 hours per day and were allowed to eat all they wanted, ate in proportion to their caloric needs. Rats that exercised for less than an hour per day consumed calories at a rate inversely proportional to their level of activity. Thus, the health and longevity of sedentary lab rats might be expected to increase if their diets were restricted. This need not be true of rats 鈥� or, by extension, humans 鈥� that are able or required to exercise.
Sudbury, Massachusetts, US
Distracting shadows
Having found that autistic children are confused by shadows as part of drawn objects, Umberto Castiello concludes that they are distracted by shadows, and suggests that classrooms should therefore be well-lit, according to your report (22 May, p 16).
It seems to me that what his experiment detected was distractions within a task, not shadows outside the task 鈥� and that autistic children take more time to process information, whether that information is relevant or contradictory.
This suggests that autistic children are unable to quickly decide the relevance of information within the task in hand. On the plus side, they may be better at noticing potential contradictions which neurotypical children may ignore.
Given this hypothesis, teaching materials used by autistic children should either exclude irrelevant information, or tell children to ignore it. For example, a maths textbook that prints its equations in blue should say that the colour is irrelevant to the meaning and is just there to make the equations easier to find.
Similarly, if all the examples for a mathematical technique use even numbers, an autistic child might assume the technique only works for even numbers 鈥� because “even numbers” is implicitly part of the information given.
That's life
I was a little nettled when I read the original article on biodiversity (24 April, p 32), because there was no mention at all of the greatest part of the biodiversity on this planet: viruses. There are more viruses on Earth than any other kind of organism, and virus genomes provide the greatest source of gene diversity 鈥� yet they don’t rate a mention. Then, on a subsequent Letters page, people take up cudgels on behalf of fungi, of all things (22 May, p 28). Cellism, that’s what it is…
Tell the truth
As an engineer and scientist, I was dismayed to read the article in which Bob Ward suggests that the scientific community ought to use public-relations techniques to rectify society’s view of science (29 May, p 26). If a few scientists use unscientific methods to advance their views then we should denounce them, not protect them. PR, like any propaganda, is used to cover the truth and not to advance it. PR is used by commercial firms and politicians when they want to rescue their public image. As scientists, our best way of gaining the public’s support is to always adhere to the truth.
iArgue
The free iPhone /iPad application Skeptical Science contains a comprehensive set of counter-arguments to climate-deniers, one of the groups mentioned in your special on denial (15 May, p 35). The app gives detailed rebuttals to more than 100 different arguments a denier may use 鈥� very useful in the pub when you don’t have the scientific evidence to hand. It is unlikely to convert an obstinate denier, but may at least sow a seed of doubt, leaning them towards scepticism.
For the record
鈥� Paul Tetlock works at Columbia University in New York, not Yale University as we stated (19 June, p 20).
鈥� In the diagram showing the monarch butterfly’s migration, we illustrated a migratory pathway across the Gulf of Mexico from Florida to Mexico (5 June, p 33). In fact, this pathway has been proposed but not confirmed 鈥� and would represent a minor route compared with the major migratory pathway through Texas.