I am thinking of a variation of the dice game, where
one has the option to throw a die unlimited number of times.The first throw is free and every next throw costs 1 dollar. One will earn the face value of the die and has the option to stop after each throw and walk away with the money earned. The earnings are not additive. What is the expected payoff of this game?
If I calculate the expected value as
for 2 rolls as (1/6)(3+4+5+6) + (1/3) (3.5-1) = 3.83$ for 3 rolls as (1/6)(3+4+5+6) + (1/3) (3.83-1) = 3.94$ for 4 rolls as (1/6)(3+4+5+6) + (1/3) (3.94-1) = 3.98$ for 5 rolls as (1/6)(3+4+5+6) + (1/3) (3.98-1) = 3.99$
it asymptotically tends to 4
Is this approach correct?
Shoudn't the player go bankrupt after 7 rolls (negative expectation)?