(class was delayed because of a firealarm, so this lecture was much sorter than usual).
Cryptogram testing for English
Lwi khqqhmi, xwduw hl tdbql quhbuigo hpcdllip vq,
avduzgo mbix yhbbnxib hyp gnxib; xi xibi hgcnql riyl
pnvrgi; oil qldgg xi kibqdqlip dy chzdym nvb xho
lwbnvmw dl. Hl giymlw xi iylibip h xdpib qkhui, hyp lwi
gnx bnnt widmwliyip; rvl, hq xi unymbhlvghlip nvbqigeiq
ny lwdq uwhymi, nvb lnbuw xhq ijldymvdqwip ro h uvbbiyl
nt hdb, hyp xi xibi gitl dy vllib phbzyiqq. Lwi mvdpiq
rbdym xdlw lwic chlibdhgq tnb biyixdym lwi gdmwl, rvl
xi whp ynyinvb nygo biqnvbui xhq ln bilvby hq xi
uhci. Xi mbnkip bnvyp lwi xdpiyip qkhui ln tdyp lwi
iylbhyui, hyp htlib h ldci thyudip lwhl xi whp
qvuuiipip. Lwdq kbneip wnxieib ln ri h qiunyp khqqhmi,
xwduw iedpiylgo hquiypip. Dl libcdyhlip gdzi lwi
tnbcib; lwnvmw qncilwdym hkkbnhuwdym ln h bho, xi unvgp
ynl ligg xwiyui, qwip h eibo pnvrltvg lxdgdmwl dy lwi
qkhui. Ro pimbiiq, nvb ioiq mbix qncixwhl huuvqlncip ln
lwdq pdcyiqq, hyp xi kibuideip lwhl lwibi xhq yn pdbiul
khqqhmi gihpdym vq tvblwib; rvl lwhl dl xhq knqqdrgi ln
ugdcr nyi qdpi nt lwi uheiby ln h gnx hbuw hl lnk,
xwduw kbncdqip h cnbi ihqo khlw, tbnc xwiyui xi ynx
pdquneibip lwhl lwdq gdmwl kbnuiipip. Xdlw unyqdpibhrgi
pdttduvglo xi qubhcrgip vk, hyp uhci ln hynlwib khqqhmi
xdlw qldgg cnbi nt dggvcdyhldny, hyp lwdq gip ln
has plain text
The passage, which at first scarcely admitted us,
quickly grew narrower and lower; we were almost bent
double; yet still we persisted in making our way
through it. At length we entered a wider space, and the
low roof heightened; but, as we congratulated ourselves
on this change, our torch was extinguished by a current
of air, and we were left in utter darkness. The guides
bring with them materials for renewing the light, but
we had noneour only resource was to return as we
came. We groped round the widened space to find the
entrance, and after a time fancied that we had
succeeded. This proved however to be a second passage,
which evidently ascended. It terminated like the
former; though something approaching to a ray, we could
not tell whence, shed a very doubtful twilight in the
space. By degrees, our eyes grew somewhat accustomed to
this dimness, and we perceived that there was no direct
passage leading us further; but that it was possible to
climb one side of the cavern to a low arch at top,
which promised a more easy path, from whence we now
discovered that this light proceeded. With considerable
difficulty we scrambled up, and came to another passage
with still more of illumination, and this led to
another ascent like the former.
Combat in the game of risk
A 1/2 inning of baseball
Introduce digraph representation of probabilities of changing from 1 state to the next.
In Oz, the weather switches among 3 states every day... Sunny, Rainy, or Snowy. And given the state of the weather today, we know how to predict the state of the weather tomorrow.
Prob( x > y )  Sunny today  Rainy today  Snowy today 

Sunny tomorrow  1/2  1/2  1/4 
Rainy tomorrow  1/4 

1/4 
Snowy tomorrow  1/4  1/2  1/2 
We can also represent the transitions between states with a labelled directed graph.
Show how learn composition can be rewritten as \(p(t+1) = A p(t)\) where
\[A := \frac{1}{4} \begin{bmatrix} 2 & 2 & 1 \\ 1 & 0 & 1\\ 1 & 2 & 2\end{bmatrix}\]