If we have three coins. Then the probability of getting three heads or three tails is:

0.5^3 + 0.5^3 = 0.25

Now, what is the probability of getting three heads or three tails when the three
coins are correlated with correlation coefficient p ??

I noticed that if I use add instead on using an integral int I dont get the same value. why ??

restart:
E := 7:   p := .5:
add(p^n*2^(n-1), n = 1 .. E)-p^E*add(2^(n-1), n = 1 .. E);
evalf(int(p^n*2^(n-1), n = 1 .. E)-p^E*(int(2^(n-1), n = 1 .. E)), 5);
 
                                  2.5078

I was reading about Thomas M. Cover Universal portfolios.
earthchangesmedia.com/breaking/April2000/0418universal.htm

One of the contributing factors seem to be related to exponential growth.
Two sentences caughts my eye:

"With the universal portfolio algorithm, profits grows exponentially.
The average of exponential growth rates has the same growth rate as the maximum"

I found the below text here :  scienceblogs.com/goodmath/2008/07/back_to_math_solving_zerosum_g.php

My question is how can I set up this problem in Maple ?!

The objective function should be: min(E(H1), E(H2), E(H3))
constraint:  Σp_i=1, and ∀i: 0≤p_i≤1.
solution: p₁=0.57, p₂=0.17, p₃=0.26.

I have this coin toss (see attached pdf and maple worksheet for further details)

"A fair coin is tossed, and the result is shown to player one. Player one must then decide
whether to pass or play. If player one passes, then he must pay player two $1. If player
one plays, then player two (who does not know the result of the coin toss) may either pass or play.
If player two pass, then he pays player one $1. If player two plays and the coin comes up heads,
then he pays player one $2. If player two plays and the coin comes up tails, then player one
must pay him $2." 
 

The objective is to determind the optimal strategy for player1. If we defined:

Action2- Play on heads and Play on tails

Action4 - Play on heads and Pass on tails

Then player1 should select Action4 67% of the time and Action2 33% of the time.

The problem is that I cant confirm such result though simulation. Does anyone have any code that works ?

Download 8342_Coin Tossing with Bluffing (winston).pdf
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View 8342_A Coin Toss Game with Bluffing-2.mw on MapleNet or Download 8342_A Coin Toss Game with Bluffing-2.mw
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