Transcript day17

Stat 35b: Introduction to Probability with Applications to Poker
Outline for the day:
1.
2.
3.
4.
5.
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, hershey’s 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 
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u
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%.
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%.
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/√n) ---> Standard Normal. (mean 0, SD 1).
In other words, X has mean µ and a standard deviation of s÷√n.
As n increases, (s ÷ √n) 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(X+Y) = E(X) + E(Y),
and V(X+Y) = 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) > V(X).
So, additional bets increase the variance of your total!
5. CLT and pairs.
Central Limit Theorem (CLT): if X1 , X2 …, Xn are iid with mean µ& SD s, then
(X - µ) ÷ (s/√n) ---> Standard Normal. (mean 0, SD 1).
In other words, X has mean µ and a standard deviation of s÷√n.
Two interesting things about this:
(i) As n --> ∞. X --> normal.
e.g. average number of pairs per hand, out of n hands.
µ = P(pair) = 3/51 = 5.88%.
(ii) About 95% of the time, a std normal random variable is within -2 to +2.
So 95% of the time, (X - µ) ÷ (s/√n) is within -2 to +2.
So 95% of the time, (X - µ) is within -2 (s/√n) to +2 (s/√n).
So 95% of the time, X is within µ - 2 (s/√n) to µ + 2 (s/√n).
That is, 95% of the time, X is in the interval µ +/- 2 (s/√n).