Thursday, February 01, 2007

Conditional Proability

Here is an example of conditional probability from real life. I have found the confusion addressed here in many minds.

Scenario: A couple has a child already. Call it a girl. If they are pregnant again, many people assume its most likely to be a boy this time as overall there has to be balance. In fact if you see around don't you see a mix more often than two boys? yes. And than two girls? yes again.

Actually, the second child is not really a dependent trial (on the first one). Again this time its as likely to be a boy as it is to be a girl. So what trips the conventional logic in the above is that though the mix is more likely than either of "two boys" and "two girls" when seen as same sex versus different sexes the outcomes are equally likely.

To analyze in a more explicit way: Possible child sex combinations are BB,BG,GB,GG. Now if the first child is already a boy the probabilities of each of those outcomes is 0,0,0.5,0.5 although before the first child was born it was 0.25,0.25,0.25,0.25.

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