tag:blogger.com,1999:blog-6894866515532737257.post4017272812034612144..comments2024-09-18T04:06:40.413-07:00Comments on Probably Overthinking It: Somebody bet on the BayesAllen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6894866515532737257.post-63193460916016044472011-11-03T10:46:10.303-07:002011-11-03T10:46:10.303-07:00That's exactly what happened to me. I did it t...That's exactly what happened to me. I did it the way you did first then decided based on the relatively simple answer that there must be another way.Ted Bunnhttps://www.blogger.com/profile/12230509214302717664noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-69176094722834492762011-11-03T10:40:26.417-07:002011-11-03T10:40:26.417-07:00@Ted: very nice. Thanks!
When I did the algebra ...@Ted: very nice. Thanks!<br /><br />When I did the algebra and everything worked out so neatly, I should have recognized that a symptom that there is an easier way to get there.Allen Downeyhttps://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-71762056833064475202011-11-03T10:17:45.462-07:002011-11-03T10:17:45.462-07:00I found it a little easier to analyze this situati...I found it a little easier to analyze this situation by realizing that you can calculate the probability of each parent's readheadedness separately. The prior probabilities relating to the two parents are independent, and the evidence "factors" into a statement just about the mother (mother passed on "a" to child) and a statement just about the father (father passed on "a" to child). So the posterior probability for the various genetic makeups of the mother and father are independent, and you can calculate them one at a time.<br /><br />Consider just the mother. The prior probabilities for her genetic makeup are what you said:<br /><br />p(AA)=q^2<br />P(Aa)=2pq<br />P(aa)=p^2<br /><br />The evidence E is that the mother passed on a to her child. <br /><br />P(E | AA) = 0<br />P(E | Aa) = 1/2<br />P(E | aa) = 1<br /><br />Turning the Bayes's theorem crank, we get<br /><br />P(AA | E) = 0 (of course)<br />P(Aa | E) = q<br />P(aa | E) = p<br /><br />So the probability that the mother didn't have red hair is q. The same reasoning works for the father, so the probability that neither had red hair is q^2.Ted Bunnhttps://www.blogger.com/profile/12230509214302717664noreply@blogger.com