# What are the probabilities of getting any or all five cards of Exodia using a specified monster card effect

In Yu-Gi-Oh!, there exists a monster effect card called Morphing Jar, which has the following effect:

FLIP: Both players discard their entire hands, then draw 5 cards.

There are more (magic/trap/monster effect) cards that allow the duelist/player to draw a defined number of cards.

As an extension of my previous question, I'm looking for the probabilities of getting any or all five cards of Exodia using the quoted effect above.

• the probability is 0 as morphing jar is banned Dec 18 '17 at 8:16

This questions is substantially different from the first version. Given the mechanics of Exodia, each of the five parts must be collected in hand. To support that, 0 Exodia cards can be drawn prior to the use of Morphing Jar. In this, the situation is similar to the linked question with the emphasized difference.

# 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 = 0

# 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 After using Morphing Jar and with subsequent draw Phases. 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). 0 Exodia cards can be present in those x cards, because Morphing Jar forces the hand to be discarded. Per the image, the chances of drawing zero Exodia cards decays the longer the games go on. When the player uses Morphing Jar, assuming he or she has drawn zero Exodia cards, he or she has a chance to draw zero to five Exodia cards with the effect. While the math is similar to the linked question, each is a distinct case.

d = the number of cards drawn so far = range (6, 35)
x = the number of Exodia cards drawn with Morphing Jar = range (0, 5)
n = the number of cards drawn by Morphing Jar = 5

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

Again, these individual cases can be evaluated with Excel, to minimize the tedious nature of each combinatoric equation.  These images portray the probability the player will draw, zero through five, Exodia cards upon playing Morphing Jar, dependent on how many cards he or she has seen previously. The first graphs the data, the second is the raw data. The sum of the probabilities for each case, for any number of cards previously seen, is 100%.

# The Rest of the Set

After playing Morphing Jar, the process for determining the probability for drawing the Exodia cards not seen is the same as described in the linked question.

d = number of cards drawn so far = range (6, 40)
e = number of Exodia cards in hand = range (0, 5)
N = number of items in the population = number of cards remaining in the deck = 40 - d
k: The number of items in the population that are classified as successes. = number of Exodia Cards in the deck = 5 - e
n: The number of items in the sample = Variable through experimentation = range (1, N)
x: The number of items in the sample that are classified as successes. = Number of Exodia Cards that must be drawn = 5 - e

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