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There are three different and independent types of buttons.

There is x number of total buttons, made up by a random combination of these three different types of buttons. Every time you press a button, you get 5 cents, however buttons have a probability of deactivating permanently. The first button type deactivates with a probability of 1:8, the second with 1:32 and the third with 1:128.

A button deactivating makes it so you lose all the money accumulated on that button. You can only keep the money from a particular button if you stop pressing it before it's deactivated.

The question is: for each type of button, what is the optimal number of times to press it to maximize the money we end up at the end?

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    $\begingroup$ @user2661923 A button deactivating makes it so you lose all the money accumulated on that button. You can only keep the money from a particular button if you stop pressing it before it's deactivated. $\endgroup$ Mar 8, 2023 at 16:06
  • $\begingroup$ a) What is a probability of $1:8$? Do you mean a probability of $\frac18$, i.e. $1$ in $8$? Or do you mean an odds ratio of $1:8$? This colon notation isn't usually used for probabilities. b) Why is the part about there being $x$ buttons with a random combination of types (whose distribution you don't specify, by the way) relevant when in the end you only ask about how to optimally deal with a single button of known type? $\endgroup$
    – joriki
    Mar 8, 2023 at 17:23
  • $\begingroup$ @joriki The distribution shouldn’t be relevant. The way I see it, optimizing for each button individually should optimize an distribution of buttons. Also, 1:8 is a 1/8 probability. $\endgroup$ Mar 8, 2023 at 20:12

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