In-class Problem Week 14, Mon.

"
Problem 3. There were n Immortal Warriors born into our world, but in the end there can be
only one. The Immortals' original plan was to stalk the world for centuries, dueling one another
with ancient swords in dramatic landscapes until only one survivor remained. However, after a
thought-provoking discussion of probabilistic independence, they opt to give the following protocol ....
"

We can ask same question again, but with the change that

If exactly _two_ Immortal flips heads, then they are declared The Pair.


b) 
What is the probability that the experiment succeeds as a function of p and n?


n(n-1)/2 * p^2 (1-p)^{n-2}

c)
How should p, the bias of the coin, be chosen in order to maximize the probability that the
experiment succeeds? (You¡¯re going to have to compute a derivative!)


By differentiation again, we can find the probability is 
maximzed when

p=2/n

d) 
What is the probability of success if p is chosen in this way? What quantity does this approach
when n, the number of Immortal Warriors, grows large?


Success probability:

n(n-1)/2  (2/n)^2 (1-2/n) ^{n-2}

In the limit, this tends to 

2 e^{-2}

~= 0.27067