Ex. Flip a coin:

When the first head (H) comes up, I quit the game

How many flips, on average, does it take to quit the game?

formula: (conditional expectation)

E(x)=E(xY=y)P(Y=y)

->E(x)=E(xY=H)P(Y=H)+E(xY=T)P(Y=T)

->E(x)=11/3+(1+E(x))2/3

I can solve from here.

Compare to other example:

Ex. Random market

Starting with initial asset value = $1

How long does it take to go broke? (asset value = 0)

-> probability 1/2 that I gain $1 = probability 1/2 that I lose $1

use same formula:

E(x)=E(xY=y)P(Y=y)

->E(x)=E(xY=+1)P(Y=+1)+E(xY=1)P(Y=1)

->E(x)=(1+2E(x))1/2+11/2

I can solve from here.

Questions:

For the parts in red - why is example 2 (1+2E(x)) and not(1+E(x)) ??

What is the difference here, and what other types of examples would it change to (1+3E(x)) . . . etc. ?