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My question relates to how behavior can change win rate in a game

Imagine a game that gives a bonus for the first win of the day. A behavior may arise where a player will play a series of games each day until they win for the first time. This earns them the bonus for that day. So, if you win the first game, you are done. You have earned the bonus. If you lose, you keep playing additional games until you win and get the bonus, then you quit for the day.

Assuming you have a 50% chance to win any individual game, will your overall win rate be less than 50% after many days of play?

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    $\begingroup$ A real world application of this is sex ratio. In many cultures it is desirable to have a boy to carry on the family name. Imagine there are no abortions and for each single birth a boy and girl are equally likely outcomes. Imagine further that every family has children until they have a boy, and then they stop. You still up with a 50:50 ratio. Some families have no girls but some have a lot of girls. Each family has a boy. It doesn't really matter how the parents decide how many kids below, each birth is still a 50:50 gamble on boy or girl. +1 to Joriki for the math. $\endgroup$
    – Readin
    Jun 6, 2016 at 6:06

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No. The behaviour induced by the bonus only affects the distribution of the wins and losses over the days, not the overall win rate. This is true quite generally, but we can also check it explicitly in this case. If you have probability $p$ to win any given game, then on any given day you'll play $k$ games with probability $p(1-p)^{k-1}$. The expected number of games won per day is $1$, and the expected number of games played per day is

$$ \sum_{k=0}^\infty kp(1-p)^{k-1}=\frac1p\;, $$

so you're still winning a fraction $p$ of the games.

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