DAPC24-067.Expected Error

You are typing your password at a sudo prompt, but suddenly, one of your fingers slipped onto a wrong key. Because the terminal hides the characters that you type, you are uncertain whether you have typed an extra character. To finish typing your password, you consider three possible strategies.

1. Continue to type the rest of your password.
2. Press backspace to delete the last character and type the rest of your password.
3. Restart typing your password from scratch.

To determine the optimal strategy, make use of the following typing speed assumptions.

• Typing any character of your password takes $0.1$ seconds.

• Pressing backspace or submitting your password also takes $0.1$ seconds.

• To restart typing your password, you delete all characters, which takes $0.3$ seconds.

• If you submit the wrong password, it takes an additional $0.3$ seconds to realize and start typing your password at a new, empty prompt.

You are given the number of characters in your password $n$, the number of correctly typed characters $k$ before your finger slipped, and a probability of $p\%$ indicating the likelihood that you pressed a wrong key and ended up with $k+1$ characters. Assuming you make no further errors, determine which strategy yields the lowest expected time to finish typing your password.