Half-Kelly Betting With Equal Wins and Losses
You start with $\
$ and make 6 consecutive bets on independent fair coin flips. Each time, you bet exactly half of your current bankroll: if you win, you gain your stake (your bankroll is multiplied by $3/2$); if you lose, you lose your stake (your bankroll is multiplied by /2$).
Suppose that across the 6 flips you win exactly 3 times and lose exactly 3 times. What is your final bankroll, and what is your net profit or loss?
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