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A statistical quandary

This post takes me back to my favourite subject of statistics

We’ve all heard the “lies, damn lies and statistics” quote and still, all too frequently, statistics have been made “to fit” a required outcome

Most people know that this is wrong but still blindly accept the results, probably from a lack of their own self-confidence to question the results

The point still remains the same and the goal is unchanged; we want to better understand and predict the world around us and mathematics offers us the clearest path to that understanding What’s lacking, quite often, is the right information required to correctly analyse a situation and come to a correct answer

Probability 101

Q. I have two children and one is a boy, what is the probability that I have boys?

Common sense tells me that the other child has (for the purposes of this experiment) a 50/50 chance of being either gender so the the common sense answer would be 1/2. Except that this is not true as is has a precedent (I already have one boy). The possible combinations of children are BG, GB, BB or GG and since I already have one boy this removes GG from the equation leaving the probability as 1/3 of having two boys.

Probability 102

Q. I have two children and one is boy born on Tuesday. What is the probability that I have two boys?

Again, common sense suggests that it will be the same as above, why would the day of birth make any difference to the statistical outcome. But it does.

Let’s, using the above naming convention, call a boy born on a Tuesday a BTu. This gives the following scenarios.

* When the first child is a BTu and the second is girl born on any day of the week there are SEVEN possibilities.

* When the first child is a girl born on any day of the week and the second is a BTu there is an additional SEVEN possibilities.

* When the first child is a BTu and the second is a boy born on any day of the week then, again, there are SEVEN possibilities.

* Finally, there is a situation where the first child is a boy born on any day of the week and the second child is a BTu. Again there are seven possibilities but, and here it gets interesting, one of them has been counted before so there are only SIX possibilities.

Counting likely outcomes we then have a total of 7+7+7+6=27 different combinations and 13 of them include two boys the answer is 13/27, wildly different to the 18/27 (1/3) answer to the first question. This is surprisingly odd and (entertainingly) illustrates that seemingly unconnected pieces of information can make a huge and statistically very important difference to outcomes.Whilst this post is folly of sorts it does have a serious side. When you are trying to measure information to produce meaningful outcomes you really must be very careful to decide what to include and what to exclude. And, you must have a grasp of how to use the information correctly, even if the mathematics required were learned when you were 13.