Solitaire: Chance for Starting Duplicates

In Klondike Solitaire, what's the chance for starting with two face-up cards of the same rank & color (what we might call "shadows", cards which don't help us very much)?

There are 7 face-up cards in starting Solitaire, so we consider the ways to choose 7 cards randomly without any duplicates. Every time we select such a card, the number available for the next draw (avoiding "shadows") goes down by 2, while the cards in the total deck goes down by 1. Therefore the chance for no-shadows is:
$${52 \over 52} \times {50 \over 51} \times {48 \over 50} \times {46 \over 49} \times {44 \over 48} \times {42 \over 47} \times {40 \over 46} \approx 0.63$$
This in turn means that the complement event, i.e., getting any shadows at all (at least two cards with duplicate rank & color), has a probability of:
$$1 - 0.63 = 0.37 = 37 \%$$
That is: About one chance in three.

More about Solitaire. 


Facebook Learning Debacle

Yesterday, the New York times published an expose on how schools in certain (poor, rural) places are being used as guinea pigs and getting "Zucked" by the Facebook founder's "Summit Learning" initiative, yet another plan to replace teachers with computer time. The highlights are sad and predictable:
  •  A program that promises "personalized learning" at each student's individual pace. Software free of cost, but school district must buy everyone a laptop.
  • Also, Facebook/Summit collects reams of information on the students involved, and expects to keep tracking students through college and beyond. 
  • The program was built by a grand total of 5 Facebook engineers (no information on whether they have any training in education or pedagogy issues). 
  • A spokesperson says it is based on, "building a curriculum from the open internet", that is, mostly links to outside web sites. Examples given include links to the Daily Mail tabloid and Christian conversion therapy sites.
  • The program "asks schools to commit to having students meet weekly in person with teachers for at least 10 minutes", but reports say not even this is happening.
Students are reporting high levels of anxiety, eye strain, hand cramps, and even seizures. Said one parent who visited a classroom, “We’re allowing the computers to teach and the kids all looked like zombies”. Some places are seeing pushback like student protests, walkouts, and parent removing their children from schools.

One reminder from yours truly: the promise of "personalized learning" is not new. It's been around at least since multimedia in the 90's, or the PLATO computer system in the 60's, or correspondence courses in the 40's, or the Gutenberg printing press, depending on how you count such things. None of them have come close to denting the need for real human teachers.