Expected value

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In probability theory, the expected value or expectation of a random variable, intuitively, is the long-run average value of repetitions of the same experiment it represents. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity.

Drop rates are expected values, too. They are percentages that show the expected average proportion (relative frequencies) of common, rare and legendary cards in a box in the long run.

Drop rates are expected values of average relative frequencies.

Definition

Let $X$ be a random variable with a finite number of finite outcomes $x_1$, $x_2$, ..., $x_k$ occurring with probabilities $p_1$, $p_2$, ..., $p_k$, respectively. The expected value of $X$ is defined as

$\operatorname{E}[X] = \sum_{i=1}^k x_i\, p_i = x_1p_1 + x_2p_2 + \ldots + x_kp_k$.

Since all probabilities $p_i$ add up to 1 ($p_1 + p_2 + \ldots + p_k = 1$), the expected value is the weighted average, with $p_i$’s being the weights.

In the special case of all probabilities being the same ($p_1 = p_2 = \ldots = p_k$), the weighted average turns into the simple average. In this case (all outcomes of $X$ have the same probability $p$), the general formula becomes

$\operatorname{E}[X] = \sum_{i=1}^k x_i\, p = x_1p + x_2p + \ldots + x_kp$.

Examples

Simple average

In the Asphalt games, simple averages with equal probabilites for all outcomes are very rare. An example could be the Elite Supplies box of the McLaren 720S World Tour. It has a guarantee to grant 1 to 5 blueprints for the car, but the actual amount of blueprints in one box is random. Although not officially stated, this example assumes that the probabilites for all amounts of blueprints are equal.

If $B$ represents the amount of blueprints obtained from opening an Elite Supplies box, the possible values for $B$ are 1, 2, 3, 4, and 5, all equally likely (each having the probability of $\tfrac{1}{ 5 }$). The expectation of $B$ is

$\operatorname{E}[B] = 1\cdot\frac15 + 2\cdot\frac15 + 3\cdot\frac15 + 4\cdot\frac15 + 5\cdot\frac15 = 3$.

This means that in the long run, a player can expect 3 blueprints per box.

The official box info, in this case, is misleading as it states a drop rate of 14.29 % for "Blueprint Boxes". Contrary to intuitive perception, this means that every Elite Supplies box contains a sub-box with 1-5 blueprints—a "box inside the box". The drop rate only states that this sub-box will make up 1 of the 7 items in the Elite Supplies box, thus $\tfrac17 = 14.29 \%$. However, it says nothing about the content of the sub-box.

The sub-box principle is a genuine feature of Festival Bundles and World Tour Supplies and illustrates a general dilemma resulting from Gameloft not labelling box info values precisely: The term "Blueprint Boxes" refers to the fact that the content of the sub-box comes as separate 1-blueprint-boxes if the player's inventory is full, whereas the drop rates of the main box refer to the items in it, with the blueprint sub-box being one of the items.

In mathematical terms, Bundles and Supplies are composed experiments consisting of several different sub-experiments, all of them with different sample spaces:

• The official drop rates show the relative frequencies of the sub-experiments, not of cards.
• One of the sub-experiments is the blueprint sub-box. It is a 1-trial experiment with the sample space $\{1, 2, \ldots, n\}$, where $n$ denotes the maximum number of blueprint cards a player can get.

Weighted average

Almost every random process in Asphalt has weighted probabilities, but they are not communicated. This is part of the game. The rarity of Pro Kit cards which divides them into common, rare and legendary can only provide a rough orientation.

On the other hand, every process of granting a random item in the game can be described as a Bernoulli trial with weighted probabilities—the player either gets a desired item or not. If $D$ represents the amount of desired items obtained from such a process (like revealing a Pro Kit card or watching an ad), the possible values for $D$ are only 1 (success, probability $p$) or 0 (failure, probability $1-p$). The expectation of $D$ is always

$\operatorname{E}[D] = 1\cdot p + 0\cdot (1-p) = p$.

This reflects Borel's law of large numbers which asserts that for independent repetitions of an identical Bernoulli trial, the expected value of the average relative frequency of an event (= its drop rate) almost surely converges to the probability of the event's occurrence on any particular trial.

This is the reason why an official drop rate often equals the probability of getting a desired item, but not always: As soon as there are additional rules for some of the cards in the box, the Bernoulli trials for each card are no longer identical. The Extra Fusion Box †, for example, guarantees that 1 of its 4 cards will be rare, so revealing this card will deliver a rare card with a probability of 100 % instead of the 34.69 % given in the box info.

See also

• Drop rate
• Probability
• Law of large numbers