Fun with probability and dice

My 3rd grader's class is doing money, fractions, and probability, and I've been thinking up a brief parent volunteer lesson plan to talk about probability with dice, around the idea of having two people each roll a die with a different number of sides, and talking about how likely it is that one or the other roll the higher number.

I'd start with d4 vs. d6; there are 24 possibilities and it's easy enough to work with the kids to enumerate all of them.  There are 14 wins for the d6, 6 for the d4, and 4 ties.

d6 vs. d8 is tougher to do in a short lesson; we'll work out that there are 48 possible outcomes, and there are 6 ties in those outcomes.  Maybe we'll have time to work out that the number of wins for the d6 is 1+2+3+4+5, or the sum of number from 1 to (number of sides-1), and therefore the number of wins for d8 is 48 - 6 - 15 = 27, but that's probably too much to get through in a single day. 

So we're all set, except... in d4 vs. d6, 14 - 6 = 8 = 2 * 4; in d6 vs. d8, 27 - 15 = 12 = 2 * 6.  Hunh.

Dice          Larger  Smaller  Ties
              Wins    Wins
d4  vs. d6    14      6        4
d6  vs. d8    27      15       6
d8  vs. d10   44      28       8
d10 vs. d12   65      45       10
d12 vs. d20   162     66       12

The difference in the number of wins is the number of sides on the smaller die times the difference in the size of the dice.  I'd never noticed this before.  Cool.

The visual explanation for this relationship is as follows (looking at the possible outcomes of d4 vs. d6):

1,1  2,1  3,1  4,1
1,2  2,2  3,2  4,2
1,3  2,3  3,3  4,3
1,4  2,4  3,4  4,4
1,5  2,5  3,5  4,5
1,6  2,6  3,6  4,6

The red area above the diagonal shows where the smaller die wins, the blue on the diagonal shows the ties, and the green below the diagonal shows where the larger die wins.  In the first four rows, the numbers of wins are balanced, but once past the possibility of a tie, the larger die wins every possible outcome in the row.  The number of outcomes in each row is the size of the smaller die, and the number of rows in which the the larger die wins all possible outcomes is the difference in the sizes of the dice.

Ah, of course.  It looks obvious from this angle, but wasn't immediately so coming at it from the other side.

Shades of Milk and Honey; Mary Robinette Kowal; 2010

Through no fault of its own (having been published a full year earlier in 2010), it is difficult to remember that the title of this book is not, in fact, 50 Shades of Milk and Honey, though I am sure there would be a market for Regency BDSM.  It is, as advertised, a fun comedy of manners against the backdrop of the Napoleonic wars.  Oh, and there's magic, though it's somewhat different from what is presented in Jonathan Strange & Mr Norrell. 

So, go and read it.  It's a very fast 300 pages.

thar be spoilers

Given Beth's history, and what we see of her in this book, I don't quite believe that she's still so impressionable and gullible, and so the "reveal" of Dunkirk's character flaws, combined with Jane's sudden interest in Mr. Vincent -- there really isn't a spark between them, so it reads a little as if she only falls in love with Mr. Vincent because he has fallen in love with her (shades of the Princess Frog) -- is not really satisfying.  These are minor blemishes on; it's still a good read.

But at any rate, this book's dark secret, likely unknown even to the author, is that it's the greatest piece of Hermione/Snape fan fiction written to date**.  To be sure, the names are changed, but who else could the plain but talented Jane and the misunderstood misanthrope Mr. Vincent be?   Ron (as Mr. Dunkirk) even makes an appearance, along with his impressionable-but-bright younger sister, to provide a hurdle to the H/S romance, until Hermione figures out that Ron is too violent and doesn't really respect her opinions.  :-)

** the best music video, of course, is this one:

Talk to Me; 2007

Don Cheadle stars as real life convict-turned-DJ Petey Greene.  

The first act of the movie, tracing Petey's rise from prison to a leading voice in D.C.'s radio scene, is excellent, and about as good a biopic as you could ask for -- I especially liked the cuts that provide links between Petey and Dewey, and the prison inmates and the protestors on the D.C. Mall. 

The problem is that, after Petey's coverage of the death of Martin Luther King, Jr., the filmmakers struggle to manufacture second and third acts around the fracturing and eventual mending of Dewey's and Petey's relationship.  Petey's failure to perform on the Tonight Show and Dewey's eulogy at Petey's funeral, aside from being entirely fictional events, simply don't carry the same emotional power as the events surrounding King's death, so the latter parts of the film drag.

The solution, I think, is to start wrapping up the film shortly after King's death, at the emotional high point.  It would be a good challenge for a film student, but you could probably even do this by re-cutting the existing film from 120 minutes to 90 minutes.

My spiral notebook was the envy of all...

The subject is TPK.  Today's lecture will discuss appropriate armor choices when adventuring in flooded ruins inhabited by monstrous reptilians.  On the one hand, not wearing plate armor greatly reduces the chance of accidental drowning.  On the other hand, it greatly increases the chance of becoming lunch.

The Secret History; Donna Tartt; 1992

I finished the Secret History, but only by skimming past page 80. I couldn't get past the fact that I couldn't suspend my disbelief that any of the characters were friends with Bunny, or that he could have made it into Julian's class. It's perhaps not quite as unbelievable as Peter Pettigrew being friends with James and Sirius, but Bunny's supposed charm is a complete mystery to me.

I think the problem is that Tartt initially only shows us Bunny's awful qualities, perhaps to soften our hearts toward his classmates, and only later does she start to emphasize his good points, in order that we might feel some regret at what has happened, but at that point she's telling, not showing.