In 2004 James Surowiecki published a book with the unequivocal title The Wisdom of Crowds: Why the Many Are Smarter Than the Few and How Collective Wisdom Shapes Business, Economies, Societies and Nations. It was intended as a gloss on Charles Mackay’s 1841 book Extraordinary Popular Delusions and the Madness of Crowds. Both books are essential reading for any risk professional.

I am something of a believer in the wisdom of crowds. The other week I was fretting about the possible relegation of English Premier League soccer club West Bromwich Albion. It’s an emotional and atavistic tie for me. I always feel there is merit, as part of my overall assessment of risk, in checking online bookmakers’ odds. They surely represent the aggregated risk assessment of gamblers if nobody else. I was relieved that bookmakers were offering typically 100/1 against West Brom being relegated. My own assessment of risk is, of course, contaminated with personal anxiety so I was pleased that the crowd was more phlegmatic.

However, while I was on the online bookmaker’s website, I couldn’t help but notice that they were also accepting bets on the imminent birth of the royal baby, the next child of the Duke and Duchess of Cambridge. It struck me as weird that anyone would bet on the sex of the royal baby. Surely this was a mere coin toss, though I know that people will bet on that. Being hopelessly inquisitive I had a look. I was somewhat astonished to find these odds being offered (this was 22 April 2015, ten days before the royal birth).

Here I have used the usual formula for converting between odds and implied probabilities: odds of m / n against an event imply a probability of n / (m + n) of the event occurring. Of course, the principle of finite additivity requires that probabilities add up to one. Here they don’t and there is an overround of 7%. Like the rest of us, bookmakers have to make a living and I was unsurprised to find a Dutch book.

The odds certainly suggested that the crowd thought a girl manifestly more probable than a boy. Bookmakers shorten the odds on the outcome that is attracting the money to avoid a heavy payout on an event that the crowd seems to know something about.

Historical data on sex ratio

I started, at this stage, to doubt my assumption that boy/ girl represented no more than a coin toss, 50:50, an evens bet. As with most things, sex ratio turns out to be an interesting subject. I found this interesting research paper which showed that sex ratio was definitely dependent on factors such as the age and ethnicity of the mother. The narrative of this chart was very interesting.

However, the paper confirmed that the sex of a baby is independent of previous births, conditioned on the factors identified, and that the ratio of girls to boys is nowhere and no time greater than 1,100 to 1000, about 52% girls.

So why the odds?

Bookmakers lengthen the odds on the outcome attracting the smaller value of bets in order to encourage stakes on the less fancied outcomes, on which there is presumably less risk of having to pay out. At odds of 6/4, a punter betting £10 on a boy would receive his stake back plus £15 ( = 6 × £10 / 4 ). If we assume an equal chance of boy or girl then that is an expected return of £12.50 ( = 0.5 × £25 ) for a £10.00 stake. I’m not sure I’d seen such a good value wager since we all used to bet against Tim Henman winning Wimbledon.

Ex ante there are two superficially suggestive explanations as to the asymmetry in the odds. At least this is all my bounded rationality could imagine.

• A lot of people (mistakenly) thought that the run of five male royal births (Princes Andrew, Edward, William, Harry and George) escalated the probability of a girl being next. “It was overdue.”
• A lot of people believed that somebody “knew something” and that they knew what it was.

In his book about cognitive biases in decision making (Thinking, Fast and Slow, Allen Lane, 2011) Nobel laureate economist Daniel Kahneman describes widespread misconceptions concerning randomness of boy/ girl birth outcomes (at p115). People tend to see regularity in sequences of data as evidence of non-randomness, even where patterns are typical of, and unsurprising in, random events.

I had thought that there could not be sufficient gamblers who would be fooled by the baseless belief that a long run of boys made the next birth more likely to be a girl. But then Danny Finkelstein reminded me (The (London) Times, Saturday 25 April 2015) of a survey of UK politicians that revealed their limited ability to deal with chance and probabilities. Are politicians more or less competent with probabilities than online gamblers? That is a question for another day. I could add that the survey compared politicians of various parties but we have an on-going election campaign in the UK at the moment so I would, in the interest of balance, invite my voting-age UK readers not to draw any inferences therefrom.

The alternative is the possibility that somebody thought that somebody knew something. The parents avowed that they didn’t know. Medical staff may or may not have. The sort of people who work in VIP medicine in the UK are not the sort of people who divulge information. But one can imagine that a random shift in sentiment, perhaps because of the misconception that a girl was “overdue”, and a consequent drift in the odds, could lead others to infer that there was insight out there. It is not completely impossible. How many other situations in life and business does that model?

It’s a girl!

The wisdom of crowds or pure luck? We shall never know. I think it was Thomas Mann who observed that the best proof of the genuineness of a prophesy was that it turned out to be false. Had the royal baby been a boy we could have been sure that the crowd was mad.

To be complete, Bayes’ theorem tells us that the outcome should enhance our degree of belief in the crowd’s wisdom. But it is a modest increase (Bayes’ factor of 2, 3 deciban after Alan Turing’s suggestion) and as we were most sceptical before we remain unpersuaded.

In his book, Surowiecki identified five factors that can impair crowd intelligence. One of these is homogeneity. Insufficient diversity frustrates the inherent virtue on which the principle is founded. I wonder how much variety there is among online punters? Similarly, where judgments are made sequentially there is a danger of influence. That was surely a factor at work here. There must also have been an element of emotion, the factor that led to all those unrealistically short odds on Henman at Wimbledon on which the wise dined so well.

But I’m trusting that none of that applies to the West Brom odds.