## FANDOM

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This is part II of a series on the maths behind drop rates.

This part contains the explanations promised in part I and will show you how to extract probabilities from many percentages (drop rates) given in a box info. The boxes granted by the 5th Anniversary event will serve as examples, but the methods also work for other boxes.

## Definitions

Every box has two properties:

• the description ("Contains 4 possible rewards", "Grants 15 Engine cards! At least 1 will be an i6 card!") and
• the box info (, or for Festival Bundles) which shows a list of percentages for the items in the box.

### Probability

Sometimes the description of a box already contains information about probabilities: If an item is marked as "guaranteed" or described as "At least 1 will be [...]", the probability of getting it is 100 %.

The probability of an item is the chance of getting it in a box.

### Drop rate (expected value)

Contrary to the belief of many players, the percentages shown in the box info do not represent the probability of getting the items. Instead, they show the distribution of items that can be expected if a very large amount of boxes is opened. For example, a guaranteed blueprint in a box with 4 items has a probability of 100 % (because it's guaranteed), but its expected value in relation to all items in the box is 25 % because it will always be 1 of the 4 items, thus $\tfrac{1}{4}$ = 25 %. Box infos show expected values, not probabilities.

The drop rate or expected value of an item is its amount relative to all items in a box if a very large number of boxes is opened.

## Extracting probabilities

### Random-only boxes

Sometimes the expected value of an item allows us to deduce its probability directly. For example, if there are 4 random items in a box and 1 has an expected value of 20 % shown in the box info, it's probable that we get 20 of this item if we open 100 boxes. This number can vary because the item is still random and 100 is not a reliable size of a statistical sampling. But if we open 1,000 or even 100,000 boxes, the share of the chosen item will become more and more close to 20 %.

If there are only random items in a box, the percentage (expected value) of an item equals its probability.
Example: The Specialist Bundle of the 5th Anniversary event "contains 3 possible rewards". None of them is marked as guaranteed.

Info:

• Blueprint Boxes: 10 %
• Tokens: 3.33 %
• Credits: 23.33 %
• Pro Kit Boxes: 63.33 %

So if we open 100 boxes, we are likely to get more or less 10 blueprints.

Unfortunately, the box info doesn't say anything about the expected values of items subsumed under "Pro Kit Boxes". Apart from blueprints, tokens and credits, there are 9 other items on the list (engines, tech and class A parts). As we don't know if they have additional specific probabilities assigned to them (for example if a mid-tech card is more likely to appear than a hybrid engine), we have to assume that they all have the same probability. This means that each item counting as Pro Kit Box has a probability of $63.33 % \cdot \tfrac{1}{9} = 7.04 %$.

That's all there is to say about random-only boxes. There more boxes you open, the more probable it is that you get exactly the number of items defined by their expected values given in the info.

### Boxes with guaranteed items

As seen above, guaranteed items distort the percentages of a box. The solution is: think of them as a box inside the box!

Example: The Novice Bundle "contains 2 possible rewards". One of the items in the list (a BMW M2 SE blueprint) is marked as guaranteed.

Info:

• Blueprint Boxes: 87.5 %
• Pro Kit Boxes: 12.5 %

Now comes the "box inside the box" method. Imagine all guaranteed items as one box and the random ones as another box. Now it's clear that the first sub-box contains the guaranteed blueprint with a probability of 100 %, and the second sub-box contains a mixture of items (including a random additional blueprint) whose probabilities sum up to another 100 %. The biggest advantage is: The second sub-box is a random-only box, and we already know how to handle random-only boxes.

Now we have to get rid of the percentages of the guaranteed content. The first sub-box (with the guaranteed blueprint) makes up 50 % of the expected value, and the second box the other 50 %. These 50 % are divided into 87.5 % - 50 % = 37.5 % for the random blueprint and 12.5 % for the Pro Kit Boxes.

Expected values (= probabilities) of the random-only sub-box:

• $\tfrac{37.5}{50}$ is the same as $\tfrac{75}{100}$, so the real probability of additional random blueprints is 75 %.
• $\tfrac{12.5}{50}$ is the same as $\tfrac{25}{100}$, so the real probability of Pro Kit Boxes is 25 %.

If there aren't any additional specific probabilities assigned to the 7 items counting as Pro Kit Box, each of them has a probability of $25 % \cdot \tfrac{1}{7} = 3.57 %$.

The same goes for the Expert Bundle: It "contains 4 possible rewards", one of which is a guaranteed blueprint pack.

Info:

• Blueprint Boxes: 25 %
• Tokens: 2.5 %
• Credits: 22.5 %
• Pro Kit Boxes: 50 %

As 25 % of the items are guaranteed blueprints, the other 75 % would be divided into 2.5 % Tokens, 22.5 % Credits and 50 % Pro Kit Boxes.

Real probabilities would be:

• Tokens: $\tfrac{2.5}{75} = \tfrac{3.33}{100}$ = 3.33 %
• Credits: $\tfrac{22.5}{75} = \tfrac{30}{100}$ = 30 %
• Pro Kit Boxes: $\tfrac{50}{75} = \tfrac{66.67}{100}$ = 66.67 %

You may have noticed that I wrote "would be". That's because all my calculations had the premise that there were no hidden rules or underlying extra probabilities for certain items. But there were. When I opened my Expert Bundles, I realized that all of them had either Credits or Tokens—none of the boxes came without both. After 20 boxes it is quite certain to assume that there is a second guaranteed item which grants "currency".

Or, seen as boxes inside the box: The items "Credits" and "Tokens" have become the content of a new sub-box (which I named "Currency").

This changes the expected values:

• Blueprint Boxes: 25 % (guaranteed, like before)
• Currency: 25 % (guaranteed, Tokens and Credits)
• Pro Kit Boxes: 50 %

The new real probabilities are:

• Tokens: $\tfrac{2.5}{25} = \tfrac{10}{100}$ = 10 %
• Credits: $\tfrac{22.5}{25} = \tfrac{90}{100}$ = 90 %
• Pro Kit Boxes: $\tfrac{50}{50} = \tfrac{100}{100}$ = 100 %

If there aren't any additional specific probabilities assigned to the 13 items counting as Pro Kit Box, each of them has a probability of $100 % \cdot \tfrac{1}{13} = 7.69 %$.

The general way to extract the probabilites from boxes with guaranteed items is:

• Take away the expected values of all guaranteed items from the total of 100 % and
• set the expected values of the remaining random items in relation to the remaining percentage.

Or, as a general formula:

Let

• $D$ be a desired random item in a box,
• $G$ be all guaranteed items in this box,
• $P(D)$ be the probability of $D$ and
• $\operatorname{E}[D]$, $\operatorname{E}[G]$ be the expected values (percentages) of $D$ and $G$ as provided in the box info, then
$P(D) = \frac{\operatorname{E}[D]}{1 - \operatorname{E}[G]}$

Example:

• $D$: Pro Kit Boxes in a Novice Bundle
• $\operatorname{E}[D]$ = 12.5 % = 0.125
• $\operatorname{E}[G]$ = 50 % = 0.5 (as 1 out of 2 = 50 % is guaranteed)
$P(D) = \frac{0.125}{1 - 0.5} = 0.25 = 25 %$

This was the easy part. It explains all the values in the overview table in Part I.

Part III will show you how to calculate the probability of getting desired items if you open more than one box.

If you have questions or find mistakes: Feel free to comment!

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