Please help me solve this real-life probability problem regarding advancing in a video game

In a mobile video game, you advance the level of your weapon by using scrolls. You must use 4 scrolls for a 40% chance to successfully increase the level of your weapon by 1. If you are unsuccessful (60% chance) you lose your 4 scrolls AND the level of your weapon DECREASES by 1. Your weapon is currently at level 15. The level of your weapon will not decline below 10, even if you try to enhance it unsuccessfully at level 10 (Though unsuccessful attempts at level 10 will still use 4 scrolls).

The question is- How many scrolls should you expect on average to expend to increase the level of your weapon from 15 to 16?

#QuantitativeFinance

Easy Math Editor

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Comments

If I am reading the question correctly, I would start by considering 1 upgrade to be 4 scrolls, so that the random variable you will be considering is really just 4 times the number of upgrades. Then the question would read that there is a 40% chance than an upgrade will increase the level by 1 and that there is a 60% chance that an upgrade will decrease the level by 1; this is then a random walk problem whereby you set a variable, say $Y$ , to be the number of levels above 15, so that you are looking for the probability that you reach $Y=1$ on the proviso that $Y$ can never regress to below $-5$ .

Please clarify your problem before I proceed with a solution.

A Former Brilliant Member - 2 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$