Final answer:
The expected number of steps with positive net profit in a negatively biased gambling game is bounded by an absolute constant due to the negative expected value per round, and this holds irrespective of the total number of rounds played. This illustrates a characteristic of a random walk with drift towards losses.
Step-by-step explanation:
The question presents a scenario of a gambling model with negative expected value per round. To calculate the expected number of steps where the net profit is positive, we must understand that because the expected value per round is negative (a 2/3 chance of losing $1 versus a 1/3 chance of gaining $1), over time, the likelihood of being in profit decreases. Regardless of the number of rounds n, the probability that the net profit is positive in any given round does not depend on n and instead relates to the distribution of wins and losses.
This model can be related to the concept of a random walk with a drift to the negative side. One way to look at this problem mathematically is by noting that the expected position after n steps is -n/3 because each step contributes -1/3 on average (gaining 1 with a probability of 1/3, and losing 1 with a probability of 2/3). Thus, as n increases, the expected position becomes more negative, making the number of steps with a positive profit bounded above by some constant that does not depend on n.
To find an upper bound of the expected number of steps with a positive net profit, we might make use of inequalities or bounds that apply to such stochastic processes. A simple argument might say that if we are more likely to lose money on each step, then there must be a maximum average number of times we are ahead, no matter how many steps are made. This maximum average number would serve as our absolute constant that bounds the expected number of steps with positive profit.