Table of Contents

• Overwatch Witch Mercy Skin
• Problem
• Assumption
• Solution
• Simplification
• Result
• The Net Result

Overwatch Witch Mercy Skin

Problem

Winning Overwatch Mercy's Halloween Witch skin at the minimum cost. There are five bundle types:

Assumption

1. I can win a legendary skin every 10 boxes. There is a post (korean) that supports this assumption.
2. The chances of winning Halloween skins are drastically higher than others during the event season.
3. There are four Halloween legendary skins. By assumption 2, I have approximately 1/40 or slightly less chance of winning the witch skin for each box. (1/10 * 1/4)
4. I currently have 1100 credits, and probably I'll get more than 1900 credits after opening 50 boxes. So I will buy 50 boxes at most.

Solution

Calculate the expected cost for buying 50 or more boxes under the condition that I can stop buying bundles when I won the skin.

Here is the example:

When I buy x boxes for y won with the chance p of winning the skin

count = x
cost = y

The probability of not winning the skin f is (1-p)^x. Buy another xx boxes for yy won with the same chance p:

count = x + xx
cost = y + f*yy

And the next case of (xxx, yyy) where ff is (1-p)^xx:

count = x + xx + xxx
cost = y + f*yy + f*ff*yyy

Calculate these until the count is greater than or equal to 50.

I calculated all possible cases by brute force algorithm(including main.py).

Simplification

Ignore the order of buying bundles. Consider the following cases:

• 2, 5, 11
• 2, 11, 5
• 5, 11, 2
• …

These are all considered as 2, 5, 11. With this simplification, the algorithm becomes easier and faster.

Result

With p = 1/40, I got the following result:

34900.83, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
35051.39, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5]
35060.40, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 11, 11]
35062.75, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 11]
35245.33, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5, 5]

With p = 1/60, I got the following result:

40557.05, [2, 2, 2, 11, 11, 11, 11]
40769.93, [2, 2, 11, 11, 24]
40908.81, [2, 2, 2, 2, 2, 2, 5, 11, 11, 11]
40979.78, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 11, 11]
41110.21, [2, 24, 24]

Because buying 2-box bundles for 25 times is cumbersome I decided to follow the best one of p=1/60 case with buying 11-box bundles first.

The Net Result

I bought two 11-box bundles and won the witch skin at the 22nd try!