When a problem is computationally or mathematically difficult, a simulation can help get an approximate answer. Want to find the value of pi? Just throw a lot of (virtual) darts at a square!

The limits of light have been replaced: All matter is teleported to a random place!

How far is the average point displaced?

This rule of space should not be disregarded: You might teleport to exactly where you started.

Hint: Don't set your bar low. Use Monte-Carlo.

Give your answer to 3 decimal places.

For reference on Monte-Carlo, here is an excellent wiki.

To use the "Wait long, buy low" strategy, one buys gas once per month on the first day that the price is unlikely to be lower in a run of 30 days. If the price doesn't satisfy that condition in the first 29 days, then you buy gas on the 30th day, regardless of its price.

What is the average amount (in dollars) that you pay for gas per gallon in any given month?

**Assumptions and Details**

- For simplicity, let \(p_G\) be the Gaussian \(\displaystyle p_G(g) = \dfrac{1}{\sqrt{2\pi \sigma_G}} \exp\left[{-\frac{\left(g-\bar{g}\right)}{2\sigma_G^2}}\right]\) where \(\bar{g}=$3.00\), and \(\sigma_G = $0.25\).
- An event \(E\) is unlikely if \(p(E)<0.5\).

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