Friday, 28 January 2011

Newspaper Probability 101

The Sun had a recent article about a 'remarkable' couple whose third child was born at 7:43am. Naturally, this time isn't special in itself, but what is is that their previous 2 children were born at 7:43 as well. Crikey, that's pretty impressive, isn't it? Three children born at the same time? Well, not quite, one was born at 7:43pm, rather than am, but still, what are the chances?

The Sun reckons the couple had "defied odds estimated at 300million to one", which means it's time to play that fun tabloid game: How Do They Work That One Out?

Let's look at the situation. All three babies were born in the same minute, but what minute it was wasn't specified in advance. As such, the first baby could have been born at any time, and then what's remarkable is that the subsequent 2 were both born at that particular time as well.

The probability that a child is born in a given minute (on a 12-hour clock system) is just 1 in 720 - that's how many minutes there are in 12 hours. Next, if we assume that children are born at times completely independently of one another, then the probability that 2 will be born at a particular time is 1 in 720 x 720 = 518,400. This is way short of the 300 million the article claims, so what have they done? It's a classic mistake, they've overlooked that the first baby could have been born whenever, and so they've done 720 x 720 x 720, which is 373,248,000 - much more like the probabilistic claim being made.

So the Sun claimed 300 million, our calculations put it at a much less remarkable 518,400, and that's assuming birth times are independent (something which I cannot find data for one way or another right now). That's still a fairly long shot, but there is still the lottery ticket factor - it's fairly common for the lottery to be won by someone, even though the odds for that one person are 14 million to 1. That's because sufficiently many people by a ticket it becomes fairly likely that someone will win. In this case we need to look at how many families could experience three children born at the same time.

According to the ONS, in 2004 there were 17 million families in the UK, of which 16% had 3 or more children. That's 2.72 million families with a ticket to a 500,000 to 1 lottery. Unsurprisingly, it's not so surprising after all.

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