"Also, Siri’s agentic features - if they work as advertised - can increase Apple’s leverage over App Publishers, because now the AI - not the user - is the entity opening and clicking on the apps."
This was really interesting to me. How does one develop an app for Siri (or an AI agent in general). Is there a standard way to communicate and expose the functionality of your app?
Can you say more about this? What accounts for the difference in time spent on a pretty good vs a very good paper? Is it that you find the material in a very good paper more worthwhile, and thus are willing to expend more effort to understand it?
> There’s a pretty fuzzy line between something appropriate for the end of undergrad versus the beginning of grad school.
I agree. I've come across countless math textbooks claiming to be aimed at "beginning grad students and advanced undergrads." There's a lot of variability in that cross section of readers. Thinking back to my own senior year as an undergrad, some of my peers were extremely bright and were bound for top grad school programs, and some were just barely scraping by managing to graduate by the skin of their teeth.
You can definitely do worse than 50/50: just flip your solution and put a single white marble in one of the bags. By the way, your solution--single black marble in one bag, all other marbles in the other bag--is optimal; do you know of a way to prove it? Besides just exhaustive enumeration, that is.
> We pick one of the three coins at random, toss it 300 times and count the number of heads.
This is the procedure. One of the coins is chosen, and that same coin is flipped 300 times. Conditional on which coin was chosen, the number of heads is Binomial, so the unconditional distribution is a mixture of three Binomials.
> The sum or sample average of a simple mixture of Bernoulli distributions still converges to a Normal.
The procedure given in the example is not a sum of samples from a mixture of Bernoulli distributions. It is a mixture of sums of Bernoulli distributions.