I'm reading conflicting answers about expected value over multiple rounds.
I read this question Two different books are giving two different solutions. , and it had this question
Every day a trader either makes 50% with probability 0.6 or loses 50% with probability 0.4. What is the probability the trader will be ahead at the end of a year, 260 trading days? Over what number of days does the trader have the maximum probability of making money?
And someone in the comments calculated this as the expected value:
$(0.6⋅1.5+0.4⋅0.5)^{260}=1.1^{260}≈57,833,669,934.$
But then I go to the https://en.wikipedia.org/wiki/Kelly_criterion wikipedia page linked in this question and I see this problem:
In a study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets
And the page describes the expected value with the geometric mean returns (1.02034) not the arithmetic mean (1.04): $10,505 = {\displaystyle 25\cdot (1.02034)^{300}}$
So which answer is correct, or are they both correct saying different things?