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In Yu-Gi-Oh!, there are some cards that allow to the duelist to win a duel automatically.

In this case, I'm focusing my question on Exodia, The Forbidden One. To win a duel using Exodia, you must have the five cards (parts of Exodia) in your hand.

These cards are (in no particular order):

I'm wondering:

How to calculate the probability to get all cards of Exodia The Forbidden One based on:

  • Parts of Exodia taken from the Deck during each Draw Phase with no additional card effects.

A Yu-Gi-Oh! deck has:

  • 40 cards in total (35 monster, spell, trap cards and 5 Exodia cards).
  • There are only five Exodia cards (no copies).
  • Draw one card per turn.
  1. I posted a question on meta about this subject.
  2. The accepted answer should have a format similar to any of the linked answers.
  3. I'm looking for a detailed answer that explains how to make those calculations and the obtained results as shown in this answer.
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This image shows the probability of drawing Exodia, and how it grows for each card drawn.

The above picture reflects the probability of drawing the full set of Exodia, dependent on the number of cards drawn overall. For example, there is a 0.000152% chance the player will draw Exodia in their opening hand, and a 87.5% chance they will draw it after seeing 39 cards out of their 40 card deck. If all 40 cards are drawn, there is a 100% chance the 5 Exodia cards will be drawn.

The Statistical Theory presented in the Question is best reflected as a Hypergeometric Distribution. Applied to this problem, the variables in questions are:

Variables

N = number of items in the population = number of cards in the deck = 40
k: The number of items in the population that are classified as successes. = number of Exodia Cards in the deck = 5
n: The number of items in the sample = Variable through experimentation
x: The number of items in the sample that are classified as successes. = Number of Exodia Cards that must be drawn = 5

Narrative

There are 5 Exodia cards (k) in a deck of 40 cards (N). If the player draws all 5 Cards (x), the player wins the game. The interest of the Question is the probability of drawing all 5 cards via only the Draw Phase. The player will see five cards in the initial hand, and an additional card per turn from the draw phase, negating any external effects. From that, over the course of the game, the player will see 5 + number of turns cards in the game (x).

Equation

h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
where [aCb] = a(a - 1)(a - 2) ... (a - b + 1)/b! = a! / b!(a - b)!

By applying the Known Variables above, the player can learn the probability they will draw the complete Exodia Set, dependent on how many cards they draw overall. In this, n ranges from 5 to 40.

Since doing a tedious calculation 36 times is tiresome, computational power is a treat. The Combination formula, [aCb], is available in Microsoft Excel. Setting up a worksheet, using the equations as in this chart: Excel math

Will support the evaluation of each step in the card drawing process, providing the resultant probability:
enter image description here

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