Blog: Plutonium

Posted by Plutonium on 11th August 2012 at 11:40 (5225 Views)
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AUG
11
2012
The Perils of Probability
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Human beings' brains are not wired very well to deal with probability. Even mathematical experts sometimes get it wrong.

A common mistake among gamblers is to assume that, if there has been, say, a run of occasions when the roulette ball has stopped on a black, then the next time round it is more likely to stop on a red. Assuming there are equal numbers of black and red slots on the roulette wheel, and the wheel is not biassed in some way, next time round it is still only 50% likely to stop on a red.

If a family has 4 boys, and tries for a girl, it is still only 50% likely that they will get a girl. The gender of each child, and the colour of the roulette wheel slot the ball stops on, are both independent of other events in the series.

So if I told you that a family has two children (who are not identical twins), one of whom is a boy, what is the probability that the other child is also a boy?

Many people might say that, as the gender of one child does not influence the gender of the other (with the single exception of identical twins), then there is a 1/2 chance that the second child is also a boy.

Not so. The probability that the second child is also a boy, is actually 1/3.

The catch is that I haven't said which child (the older or the younger) is a boy.

If you take the population of families with two children, a quarter will fall into each of the following categories, where the first letter indicates the older child, and the second letter the younger child:

BB
BG
GB
GG

Now, if I say that one child is a boy, that obviously excludes the fourth category, GG. There is an equal probability that the family is in one of the other three categories. In only one of those categories is the second child a boy. Therefore, the chances of this are 1/3.

If I had said that the older child is a boy, that would exclude two categories: GB and GG. There is an equal chance that the family is in one of the remaining categories, so the chances that the second child is a boy, are 1/2. Similarly if I said that the younger child was a boy.

So one crucial bit of information changes the odds.

Some might think this is obvious, others might not. But it underpins another interesting and controversial problem in Maths, which I will come to in due course.
Updated 11th August 2012 at 11:44 by Plutonium
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  1. Scrotnig -
    AUG
    12
    2012
    Scrotnig's Avatar
    I used to bet small amounts on roulette at the casino, rarely won much though i did once get 3200 out of a fiver!

    I agree that there is a temptation to think that a colour is more likely to come up based on previous results, which isn't the case.
  2. Evil -
    AUG
    13
    2012
    Evil's Avatar
    So if I told you that a family has two children (who are not identical twins), one of whom is a boy, what is the probability that the other child is also a boy?
    I don't understand why one has to take the birth order into account.
  3. Plutonium -
    AUG
    18
    2012
    Plutonium's Avatar
    The birth order is important, because giving you information on whether the older or the younger child is a boy, changes the probability that the other child is a boy.

    Let us say that a couple have their first child. There is a 50% chance that it is a boy or a girl.

    Let's say it's a boy. The couple then have a second child. Again, there is a 50% chance that it is a boy or a girl.

    But if the first child was a girl, then, again there is a 50% chance that the second child is a boy or a girl.

    So here are the possibilities:

    1st child boy: 50% chance. 2nd child boy: 50% chance. Overall probability: 50% x 50% = 25% chance.
    1st child boy: 50% chance. 2nd child girl: 50% chance. Overall probability: 50% x 50% = 25% chance.
    1st child girl: 50% chance. 2nd child boy: 50% chance. Overall probability: 50% x 50% = 25% chance.
    1st child girl: 50% chance. 2nd child girl: 50% chance. Overall probability: 50% x 50% = 25% chance.

    So overall, the population of families with 2 children will divide fairly equally into the following four categories:

    (1) 1st child boy, 2nd child boy: 25%
    (2) 1st child boy, 2nd child girl: 25%
    (3) 1st child girl, 2nd child boy: 25%
    (4) 1st child girl, 2nd child girl: 25%

    If I tell you that the older child in a family is a boy, then the family must be in group 1 or group 2. There is a 50% of either, and therefore a 1/2 chance that the younger child is a boy.

    However, if I tell you that one child is a boy, but give no information on whether it is the older or younger child, then the family could be in group 1, 2 or 3. There is an equal chance they are in each of these, but in only one of the groups is the other child a boy. Therefore the chance of the other child being a boy is 1/3.
  4. Evil -
    AUG
    19
    2012
    Evil's Avatar
    Ahh ok, I see now. Thanks for explaining it further.

    This probability stuff is very tricky.

    Updated 19th August 2012 at 03:40 by Evil
  5. Mr Fred -
    AUG
    19
    2012
    Mr Fred's Avatar
    I guess, in all probability, most posters would have read that crap.
  6. Plutonium -
    AUG
    19
    2012
    Plutonium's Avatar
    I assume that you mean most posters would not have read that crap. Well, as I said at the outset, I'm not that bothered if ppl read my posts or not. This stuff is not everybody's cup of tea, I know that. But I like writing it, and if ppl want to read it, or don't want to read it, that is entirely up them. So comments like this just make you look like a prat, frankly.

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