tag:blogger.com,1999:blog-14840542.post6613491469496899394..comments2013-07-10T11:34:20.634-07:00Comments on The Odds Are One: Lies, Gender, and Damned StatisticsTransient Gadflyhttp://www.blogger.com/profile/10313323030838183737noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-14840542.post-58206571953829249352012-04-20T15:08:05.487-07:002012-04-20T15:08:05.487-07:00"I have two children. One of them is a boy. W..."I have two children. One of them is a boy. What are the odds the other one is also a boy?"<br /><br />First, "odds" are not the same as "probability." "Odds" are what a bookie will pay on a bet. They can be expressed as (1-P)/P:1. So if the probability is 1/2, the odds are 1:1. If the probability is 1/3, the odds are 2:1.<br /><br />Second, the answer to this question is P=1/2, not P=1/3. Martin Gardner himself retracted his answer of P=1/3 six months after posting it, but nobody seems to recall that.<br /><br />It is true that 1/3 of all 2-child families with at least one boy have two. But that is not the same condition as being told by the parent that there is a boy, because the parent of a boy and a girl is equally likely to tell about the girl, as to tell you about the boy. So, while each of the four family types {BB, BG, GB, GG} has a 1/4 chance of existing, the chances the case exists AND you will be told about a boy are 1/4, 1/8, 1/8, and 0, respectively. This makes the answer (1/4)/(1/4+1/8+1/8)=1/2.<br /><br />If you had said "I was selected to address you *because* I have two children, and one of them is a boy" then the answer is 1/3. But that is not what you asked. The difference is whether "one boy" is a requirement for selection, or an observation after selection.<br /><br />"I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?"<br /><br />Also 1/2. You can get this answer either intuitively, or by writing out your table *AND* letting the parent of different types choose what to tell us randomly. The reason people object to your answer changing from 1/3 to 13/27, is that they view the information "born on a Tuesday" as an additional observation made after selection. Not a requirement. The reason your answer, which is based on a requirement, changes, is because a two-boy family is nearly twice as likely to have one born on a Tuesday as a one-boy family.Anonymousnoreply@blogger.com