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The templates {{Required trials}} and {{Required trials percent}} calculate the number $n$ of Bernoulli trials required to achieve at least one success with a given minimum probability.

The only difference between the two is that {{Required trials}} works with probabilities from 0 to 1, while {{Required trials percent}} works with probabilities from 0 % to 100 %.

The formula is

$n \ge \frac{\ln(1- \mathrm P_{min})}{\ln(1-p)}$
$\mathrm P_{min}$: desired minimum probability of at least one success
$p$: probability of success

## Usage

{{required trials | pmin = | p = | format}}


or

{{required trials percent | pmin = | p = | format}}


## Parameters

pmin

• Desired minimum probability $\mathrm P_{min}$ of at least one success
• Required.
• {{Required trials}}: Number between 0 and 1.
• {{Required trials percent}}: Number between 0 % and 100 %, with or without percent sign.

p

• Probability $p$ of success
• Required.
• {{Required trials}}: Number between 0 and 1.
• {{Required trials percent}}: Number between 0 % and 100 %, with or without percent sign.

format

• Display options for $n$
• Optional. One of the following texts:
• ceil – Displays the ceiling $\lceil n \rceil$ instead of an unrounded $n$.
As there are no "partial" trials, it is usually recommended to use ceil instead of round.
• round x – Rounds $n$ to x trailing digits.

## Examples

### Standard

Given a probability $p$ = 1.88 % = 0.0188 of obtaining a V8 Engine from a Pro Kit Box, the number of cards required to get at least one V8 Engine with a probability of 90 % = 0.9 is:

Unformatted
{{required trials | p=0.0188 | pmin=0.9}}121.32299622997
Rounded to 3 digits
{{required trials | p=0.0188 | pmin=0.9 | round 3}}121.323
Ceiling
{{required trials | p=0.0188 | pmin=0.9 | ceil}}122

### Percentages

Technically, {{Required trials percent}} simply calls {{Required trials}} with pmin and p divided by 100.

Percent signs and spaces are removed, so it doesn't matter if parameters are provided as 90 %, 90% or 90.

Unformatted
{{required trials percent | p=1.88 % | pmin=90 %}}121.32299622997
Rounded to 3 digits
{{required trials percent | p=1.88% | pmin=90% | round 3}}121.323
Ceiling
{{required trials percent | p=1.88 | pmin=90 | ceil}}122

### Extracted values from Pro Kit Box templates

{{Required trials percent}} is particularly useful for statistical values extracted from Pro Kit Box templates via the # parameter, as these values are always given in percent.

For example, {{Finish Line Box|#sv82}} delivers 2.13, the average percentage of V8 Engines in . See {{A8box-meta}} for an overview of statistics parameters.

Cards needed for at least one ...

V8 Engine from Finish Line Boxes
{{required trials percent | p={{Finish Line Box|#sv82}} | pmin=90 | ceil}}107
Tech card from Ultra Boxes
{{required trials percent | p={{Ultra Box|#stech2}} | pmin=90 | ceil}}37
Tech card from Engine Boxes (impossible)
{{required trials percent | p={{Engine Box 1-5|#stech2}} | pmin=90 | ceil}}$\infty$

### Extreme values

Extreme parameter values would lead to errors like division by zero or invalid arguments for the logarithm. These values are handled as follows:

p

• p = 0: success is impossible, $\infty$ is displayed (an infinite number of trials is required).
• p = 1: success is sure, 1 is displayed (only one trial is required).

pmin (if 0 < p < 1)

• pmin = 0: A desired 0 % probability of having at least one success is achievable with any probability of success, 1 is displayed (only one trial is required).
• pmin = 1: A desired 100 % probability of having at least one success is impossible if success is not sure, $\infty$ is displayed (an infinite number of trials is required).

The percent values 0 % and 100 % work accordingly.

• {{required trials | p=0 | pmin=0.9}}$\infty$
• {{required trials | p=0 | pmin=0}}$\infty$
• {{required trials | p=0 | pmin=1}}$\infty$
• {{required trials | p=1 | pmin=0.9}}1
• {{required trials | p=1 | pmin=0}}1
• {{required trials | p=1 | pmin=1}}1
• {{required trials | p=0.2 | pmin=0}}1
• {{required trials | p=0.2 | pmin=1}}$\infty$

## Technical details

If the infinity sign is displayed, the template adds an invisible leading <div> tag:

<div style="display:none;">9999999999999999999</div>$\infty$

This is to ensure that $\infty$ is sorted after all numbers in tables, for example in Pro Kit Box statistics: minimum card requirements.

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