Stat 35b: Introduction to Probability with Applications to Poker

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• Stat 35b Introduction to Probability with
  Applications to Poker
• Outline for the day
• 1. Bayes Rule again
• Gold vs. Benyamine
• Bayes Rule example
• Variance, CLT, and prop bets
• CLT and pairs
• Prizes ten 100 grand bars, pistachio nuts,
  blowpops, sunflower seeds,
• choco santas, moonpies,10 butterfingers, 10
  crunchbars, marshmallow cookies,
• Gum (2), trail mix, hersheys kissables, pez
  pencil set, 5 Lakers pens,
• peanut butter wafers, toffee popcorn, yoyos,
  spytech markers, set of erasers,
• Stapler set (mickey, winnie, cars), mint
  chapstick, play-doh,
• Clip-on calculator, peppermint candy canes, cards
  (3 batman, 1 bratz, 2 carebears).

? ? u ? ? ? u ?
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1. Bayes Rule If B1 , , Bn are disjoint
events with P(B1 or or Bn) 1, then P(Bi A)
P(A Bi ) P(Bi) ?P(A Bj)P(Bj). Ex.
Let disease mean you really have the disease,
and let mean the test says you are positive
- means the test says you are negative. Suppose
P(disease) 1, the test is 95 accurate
P( disease) 95, P(- no disease)
95 . Then what is P(disease )? Using Bayes
rule, P(disease ) P( disease)
P(disease) ------------------------------------
----------------------------------- P(
disease)P(disease) P( no disease) P(no
disease) 95
1 ---------------------------------------------
----- 95 1 5 99
16.1.
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3. Bayes rule example. Suppose P(nuts) 1,
and P(horrible hand) 10. Suppose that P(huge
bet nuts) 100, and P(huge bet horrendous
hand) 30. What is P(nuts huge bet)? P(nuts
huge bet) P(huge bet nuts)
P(nuts) ----------------------------------------
--------------------------------------------
P(huge bet nuts) P(nuts)
P(huge bet horrible hand) P(horrible hand)
100 1 ----------------------
----------------- 100 1 30
10 25.
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4. Variance, CLT, and prop bets. Central Limit
Theorem (CLT) if X1 , X2 , Xn are iid with
mean µ SD s, then (X - µ) (s/vn) ---gt
Standard Normal. (mean 0, SD 1). In other words,
X has mean µ and a standard deviation of svn.
As n increases, (s vn) decreases. So, the
more independent trials, the smaller the SD (and
variance) of X. i.e. additional bets decrease the
variance of your average. If X and Y are
independent, then E(XY) E(X) E(Y),
and V(XY) V(X) V(Y). Let X your profit
on wager 1, Y profit on wager 2. If the two
wagers are independent, then V(total profit)
V(X) V(Y) gt V(X). So, additional bets increase
the variance of your total!
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• 5. CLT and pairs.
• Central Limit Theorem (CLT) if X1 , X2 , Xn
  are iid with mean µ SD s, then
• (X - µ) (s/vn) ---gt Standard Normal. (mean
  0, SD 1).
• In other words, X has mean µ and a standard
  deviation of svn.
• Two interesting things about this
• (i) As n --gt 8. X --gt normal.
• e.g. average number of pairs per hand, out of n
  hands.
• µ P(pair) 3/51 5.88.
• About 95 of the time, a std normal random
  variable is within -2 to 2.
• So 95 of the time, (X - µ) (s/vn) is within -2
  to 2.
• So 95 of the time, (X - µ) is within -2 (s/vn)
  to 2 (s/vn).
• So 95 of the time, X is within µ - 2 (s/vn) to
  µ 2 (s/vn).
• That is, 95 of the time, X is in the interval µ
  /- 2 (s/vn).