Some argue that Beauty, upon awakening, learns no new information, and therefore the priori probabilities of the initial coin flip are valid. While it is true that B learns no new information, that is NOT what determines the posterior probabilities; they are changed by the simple act of waking up – thus entering phase two of the experiment, the first one being the coin flip. For example, consider the same rules but for flipping the coin a second time on tails to determine whether Beauty should be awoken on Monday or Tuesday. Now the answer is heads with 1/2. The equations remain the same, but Pr(C=h|D=mon) evaluates to 2/3 instead of 1/2.

(1) and (2) work for all possible scenarios with respect to awakening Beauty. For Pr(C=h|D=mon):=1 however, signifying Beauty is never awoken on Monday for tails, they evaluate to x=y, meaning Monday and heads are synonymous, thus C taking on its priori probabilities, 1/2.

]]>Let C be the coin flip {h,t}, and D the day {mon,tue}. Then

(1) y := Pr(C=h) = Pr(C=h|D=mon)*Pr(D=mon) + Pr(C=h|D=tue)*Pr(D=tue) = 1/2*x + 0*(1-x) = x/2

(2) x := Pr(D=mon) = Pr(D=mon|C=h)*y + Pr(D=mon|C=t)*Pr(C=t) = 1*y + 1/2*(1-y) = x/4 + 1/2

(1) and (2) defines an equation system in x and y with solution 2/3 and 1/3 respectively. QED.

“:=” means “defined as” – introducing a definition.

Thus, there is no need for convoluted explanations betting on outcomes, or sidetracking with modified problems. All the information needed for solving the problem at hand is readily available in the problem description.

]]>Sometimes I wonder how much the women making comments like this and advocating for these matters had actually ever felt that they are capable of earning these prizes but have experienced fear of for example, emasculating and loosing a partner or missing out on other aspects of life. Personally, that has been my experience and I’m only 23 but that has led to the delay of my undergraduate graduation and feeling pushed out of physics (and towards math, which is much more welcoming to women). The female tendency towards humility has also led to me realizing that my fellow male peers that I used to think are smarter than me are in fact not, yet they are louder and more likely to assert themselves. However, they are also more likely to pedestalize their elders in the field and, well, how do i put this in appropriate language….they’re much more likely to kiss up to authority figures with little justification. Despite what the world says, these are not characteristics of successful people and i think it’s probably more useful to point that out to promising women than to tell them that they are equally capable because often they can forget that but fundamentally they know. There’s also a need to talk about romantic relationships which are often what truly limits female careers, or sometimes, as in my case, the resulting fall out has limited my career both in terms of being able to find funds to keep going and constantly having to explain my experiences to the male professors with whom I work.

The most limiting factor of the Field’s medal in regards to women is the fact that it has an upper limit on the age of the recipient for the reasons I mentioned above (plus the extensions of those reasons later in life – ie. the effects of having children and such). Any other factors working against women are psychological and that has nothing to do with the prize. To sum up, my point of view is that men are simply more egotistical and value prizes more and women tend to value other aspects of life equally and are more likely to defer to men when they act in ways that project a false sense of confidence and superiority. It does come down to biology but not in the ways people seem to think IMO. For example, men and women can be equally good at math but women are much less likely to enjoy competing in math competitions, especially timed ones, instead preferring to take their time and enjoy the process of learning more so than the process of winning.

]]>I get that the bus number is 4, Wizard A has 2 children, of ages either 2 and 2 or 1 and 3. If the number of children were less than 4, the number of children determines their ages. If it were greater than 4, then B would be unable to determine how many children A has since 5 can be partitioned in length 2 as (3,2) or (4,1) and as length 3 as (3,1,1) or (2,2,1), appending 1s to deal with numbers greater than 5.

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