[majormajor]

> If you want Coke and you start with Pepsi, you'll get back Coke (1/2 + 1/2x1/2) or Pepsi (1/2x1/2). However, you could pick it and get back Pepsi 10 times in a row.

Why would you push it ten times in a row? If you get a Pepsi from it on your first press, you now know:

- The button labeled Pepsi is actually Random (it can't be the actual Pepsi button, since none of the labels are correct, and it's also not the Coke button)

- Therefore, the button labeled Coke is actually Pepsi (it can't be Coke by rule, and it's also not Random since we know where Random is now)

- Therefore, the button labeled Pepsi is actually Coke (elimination)

So the max is still 2 steps, but you have 75% chance of getting a Coke on the first try instead of just 50% chance. Same lower bounds, but better expected value.