User:Guy Bukzi Montag/sandbox

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Drop Rates Explained

I often see players argue about drop rates of Pro Kit Boxes. Namely during the McLaren 720S World Tour I noticed two questions that made me want to find out myself:


 * 1) Why does the box info show only 14.29 % for an item that is actually guaranteed?
 * 2) If you need V8 engines, what is the better choice: the Pro or the Elite Supplies Box?

This blog post will answer the two questions, and while we're at it, shed some light on common misconceptions about drop rates of Pro Kit Boxes in general.

Definitions
First, let's have a look at one of the McLaren 720 World Tour boxes, the Elite Supplies Box: It contains 7 items, 4 of which are guaranteed. The other 3 items are taken randomly from a list of 9 items. So it is sure that you get between 1 and 5 blueprints, credits, tokens and one class A part, but you can't be sure that you get a V8 engine, for example, because the engine is on the random list.

Drop rate
If "drop rate" is defined as "the probability of getting a specific item in a box", as a fellow racer told me, the drop rate or probability of tokens (which are guaranteed) in an Elite Supplies Box is 100 %. If you buy 100 Elite Boxes you can be sure to have tokens in every single box. To avoid misunderstandings, I'll always use the term probability from now on.


 * The drop rate of an item is the probability of getting it in a box.

Percentage
So let's look at the percentages that are shown when you tap on the icon of the Elite box. These numbers show the share of an item, whether it's guaranteed or not, in relation to the the other items inside a box.

As tokens have a probability of 100 %, they will always make up 1 of the 7 items in the box. Thus, their percentage is always $$\tfrac{1}{7}$$ = 0.142857 = 14.29 %—which is exactly the percentage provided in the box info of the game. So the question number 1 from above is already answered: The box info does not show the probability or drop rate of an item, but its percentage relative to the other items in the box.


 * The percentage of an item is its amount relative to the other items inside a box, expressed in percent.

So the notions of probability and percentage are different when an item is guaranteed to be in a box.

However, when a box only consists of items that are not guaranteed, it is possible to deduct the probability from the percentage given in the info. For example, the Random Box 2 which is rewarded in cups, shows "Legendary: 3.06 %". If you had a very large amount of these boxes, let's say 10,000, and opened them all, the amount of Legendary items you'd get would be around 306.

Unfortunately, you could not deduct that these 306 Legendary items have an equal distribution of blueprints, engines and tech cards, because we don't know if these types of cards have additional individual probabilities assigned to them. A rough approximation would be possible if one logged all types of cards whenever a box is opened. Is there anybody who did this already?

The McLaren 720S World Tour Elite Box
7 items
 * 4 guaranteed
 * 3 random, drawn from a list of 9 which contains:
 * 1 Mid-Tech
 * 1 Advanced Tech
 * 2 Early Tech
 * 2 Initial Tech
 * 2 Class A Part
 * 1 V8

What is the chance that you get a V8 engine from an Elite Box? To calculate this, we can neglect the guaranteed items. What counts is the 3 draws that are internally made from the list of the 9 random items. Technically it's an urn problem (see Wikipedia for further explanations): You have an urn with 9 items, one of which is a V8 engine. What is the chance to get this V8 engine if you draw 3 times?

There are three possibilies to get the V8: Either you get it at the first draw or the second or the third.


 * 1) V8 at first draw: The probability to get the V8 at the first draw is 1 out of 9, thus $$\tfrac{1}{9}$$. Now 8 items are left in the urn. As you already have the V8 the next draw has a probability of $$\tfrac{8}{8}$$ that you don't get a V8. Now 7 items are left, and the third draw has a probability of $$\tfrac{7}{7}$$ (that you don't get a V8). The total probability to get a V8 at the first draw is $$\tfrac{1}{9} \cdot \tfrac{8}{8} \cdot \tfrac{7}{7} = \tfrac{1}{9}$$ = 11.11 %.
 * 2) V8 at second draw: The probability that you don't get the V8 at the first draw is $$\tfrac{8}{9}$$. Then you get it with the second draw from 8 remaining items, thus the probability is $$\tfrac{1}{8}$$. Now 7 items are left, and the third draw has a probability of $$\tfrac{7}{7}$$ (that you don't get a V8). The total probability of this variant is $$\tfrac{8}{9} \cdot \tfrac{1}{8} \cdot \tfrac{7}{7} = \tfrac{1}{9}$$ = 11.11 %.
 * 3) V8 at third draw: Analogous to the previous two examples, the probability of getting the V8 at the third draw is $$\tfrac{8}{9} \cdot \tfrac{7}{8} \cdot \tfrac{1}{7} = \tfrac{1}{9}$$ = 11.11 %.

The overall probability of getting the V8 with 3 draws is the sum of the probabilities of getting it at the first, second or third draw, thus

$$\frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{3}{9} = \frac{1}{3} = 33.33 %$$.

The McLaren 720S World Tour Pro Box
4 items
 * 1 guaranteed
 * 3 random, drawn from a list of 9 which contains:
 * 1 Blueprint
 * 2 V8
 * 1 Early Tech
 * 1 Initial Tech
 * 1 Class A Part
 * 1 Advanced Tech
 * 1 Mid-Tech
 * 1 20 Tokens

This time we have the chance to get either 1 or even 2 V8s, so there are more possibilities ("0" denotes no V8, "V" a V8): V00, 0V0, 00V, VV0, V0V and 0VV.

V00: $$\tfrac{2}{9} \cdot \tfrac{7}{8} \cdot \tfrac{6}{7} = \tfrac{6}{36} = \tfrac{1}{6}$$ = 16.67 %

0V0: $$\tfrac{7}{9} \cdot \tfrac{2}{8} \cdot \tfrac{6}{7} = \tfrac{6}{36} = \tfrac{1}{6}$$ = 16.67 %

00V: $$\tfrac{7}{9} \cdot \tfrac{6}{8} \cdot \tfrac{2}{7} = \tfrac{6}{36} = \tfrac{1}{6}$$ = 16.67 %

VV0: $$\tfrac{2}{9} \cdot \tfrac{1}{8} \cdot \tfrac{7}{7} = \tfrac{1}{36}$$ = 2.78 %

V0V: $$\tfrac{2}{9} \cdot \tfrac{7}{8} \cdot \tfrac{1}{7} = \tfrac{1}{36}$$ = 2.78 %

0VV: $$\tfrac{7}{9} \cdot \tfrac{2}{8} \cdot \tfrac{1}{7} = \tfrac{1}{36}$$ = 2.78 %

The overall probability of getting exactly 1 V8 with 3 draws is the sum of the first three variants: $$\tfrac{1}{6} + \tfrac{1}{6} + \tfrac{1}{6} = \frac{3}{6} = \frac{1}{2}$$ = 50.00 %.

The overall probability of getting exactly 2 V8s is the sum of the seccond three variants: $$\tfrac{1}{36} + \tfrac{1}{36} + \tfrac{1}{36} = \frac{3}{36} = \frac{1}{12}$$ = 8.33 %.

And the probability of getting at least one V8 (i. e. 1 or 2) is the sum of the two probabilities above:

$$\frac{1}{2} + \frac{1}{12} = \frac{7}{12} = 58.33 %$$

This is the answer to question number 2: The chance of getting at least 1 V8 is significantly higher (58.33 % vs. 33.33 %) if you buy a Pro Box.

Note: These probabilities only apply if there aren't any individual probabilities assigned to the random items of the list (for example that a v8 engine doesn't have the same chance to be drawn from the urn as a Mid-Tech card).

Code appendix
For those familiar with Visual Basic for Applications: I wrote a "quick and dirty" routine to simulate the draws with a very high number of rounds to get good statistical approximations of the real numbers. If you'd like to do your own drop rate calculations for a box you can copy the code into an Excel macro and see if your they were correct.

The three McLaren 720S boxes are already contained; just comment out the two boxes you don't want to draw from and adapt the four constants at the beginning correspondingly.

I am not responsible for any problems the code may cause.