Day 15: Mario, Save Princess Peach!

Discrete Mathematics Level 4

Mario is in a sticky situation. He has one life remaining and he has one level to complete before he can save the love of his life, Princess Peach! If he beats this level he will save her, but if he doesn't, then it truly is GAME OVER.

This level is not easy and in fact he only has a \(\frac{1}{400}\) chance of beating the level.

But not all is lost. Regardless of whether he wins or loses, he has a \(\frac{1}{7}\) chance of obtaining a 1-UP in the level, which will give him an extra life; effectively another go.

What is the chance of Mario saving Princess Peach?

Your answer should be of the form \(\frac{1}{k}\) where \(k\) is an integer. Find the value of \(k\).

This problem is part of the Advent Calendar 2015.
If you can see this then well done! You might guess that I am a Mario fan, and in fact I have made a custom Super Mario World ROM on my website. Try it out if you want!

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