User:Guy Bukzi Montag/sandbox

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In my previous post I explained why the percentages shown for boxes (when you tap on the icon) don't always represent the real drop rates or probabilities that you get a certain item.

This post shows how you can deduct probabilities from these percentages in many cases. If you already have the cars offered in the 5th Anniversary event or simply don't care about them, you can still go for some engines you need, so it might be interesting how your chances are to get them.

Overview
If you're not interested in the explanations, here's the overview of all items with their real probability (drop rate) in the "P" columns. Please note that the probabilities of all items counting as "Pro Kit Boxes" are assumed as equal.

Explanations
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Questions
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Is it possible to assemble one of the cars with the coins from the event?
There are 30 quests with three levels rewarding, and. If you accomplish all quests you get a bonus reward of, so the total number of coins you can get from quests is.

Watching ads (3 times per day, average of per ad) for 10 days grants additional.

As some quests require a huge amount of finished races, it is also possible to collect about per day automatically on tracks when playing races, which would result in.

We'll take this total of as a base for further calculations.

Audi R8 e-tron Special Edition
No. Even if you got 7 blueprints from every Specialist Bundle you open, you'd need 22 boxes to reach the 150 blueprints needed to assemble the car. 22 boxes cost which is impossible to reach if you don't buy additional coins for real money.

Furthermore, the chance of getting 7 blueprints from a Specialist Bundle is $$\tfrac{1}{7}$$ = 14.29 %. However, the chance of getting 7 blueprints 22 times in a row is $$\tfrac{1}{7^{22}} = \tfrac{1}{3,909,821,048,582,988,049}$$. This is 1 out of 3.9 quintillions.

If you think of buying coins for real money: Don't. The expected (average) number of blueprints in a Specialist Bundle is
 * $$1 \cdot \tfrac{1}{7} + 2 \cdot \tfrac{1}{7} + 3 \cdot \tfrac{1}{7} + 4 \cdot \tfrac{1}{7} + 5 \cdot \tfrac{1}{7} + 6 \cdot \tfrac{1}{7} + 7 \cdot \tfrac{1}{7} = 4.$$

So you'd need 38 boxes which cost —and you still can't be sure that this is enough because you can also get less than an average of 4 blueprints per box.

Mercedes-Benz SLK 55 AMG Special Edition
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BMW M2 Special Edition
Yes. The car needs 70 blueprints to be assembled.

How many blueprints do I get for n coins?
Let's assume that you have. With those coins you can buy either
 * 3 Expert Bundles ( each) or
 * 7 Specialist Bundles ( each) or
 * 14 Novice Bundles ( each).

The Expert bundle grants at least 1 Audi R8 e-tron SE blueprint per box; the maximum number you can get is 7. Assumed that there are no underlying probabilities for certain amounts of blueprints, you can imagine this as an urn containing 7 boxes: one box with 1 blueprint, one box with 2, one with 3 and so on. If you draw one box from the urn, each box and thus each amount of blueprints has the same probability: $$\tfrac{1}{7}$$.

As you can buy 3 Expert Bundles for your coins, you can draw 3 times from the urn (with putting back the drawn item).
 * The probability of getting the maximum number of blueprints from 3 draws (i. e. 3 times box number 7) is $$\tfrac{1}{7} \cdot \tfrac{1}{7} \cdot \tfrac{1}{7} = \tfrac{1}{7^3} = \tfrac{1}{343}$$ = 0.29%.
 * Each other combination of boxes has the same probability of $$\tfrac{1}{343}$$, but there are several combinations that grant, for example, 20 blueprints (maximum number minus 1): 7-7-6, 7-6-7 and 6-7-7, each with a probability of $$\tfrac{1}{343}$$. So the probability for 20 blueprints is $$\tfrac{3}{343}$$ = 0.87%.
 * This leads to the question in how many ways you can combine three summands from 1 to 7 so that the result is the desired number of blueprints. You can imagine that as throwing a seven-sided dice three times. Technically, this is a special case of a composition problem which is examined in combinatorics (see Wikipedia article for further explanations).

The probability of getting the maximum number of items from d draws out of n items is $$\frac{1}{n^d}$$.

As the number of draws equals the number of boxes you can buy, the number of boxes is defined by $$\lfloor\frac{amount of coins}{box price}\rfloor$$ Bla

\begin{align} m &= \sum_{k=0}^{\lfloor\frac{s-n}{7}\rfloor} (-1)^k \cdot \binom nk \cdot \binom{s - 1 - 7k}{n - 1}\\ &= \sum_{k=0}^{\lfloor\frac{s-n}{7}\rfloor} (-1)^k \cdot \frac{n!}{k! \cdot (n-k)!} \cdot \binom{s - 1 - 7k}{n - 1} \end{align} $$

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with m: probability of a sum n: amount of boxes s: sum of blueprints

How many coins do I need to assemble a car with a probability of 90 %?
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Which bundle grants more tokens: Expert or Specialist?
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