- The chance of drawing all five pieces of "Exodia" in the starting hand of five cards, assuming you are using the minimum of 40 cards in your Deck, is 1 in 658,008.
5/40 * 4/39 * 3/38 * 2/37 * 1/36
- In casual play, if the banlist were to be ignored and 3 copies of every card were allowed regardless of their status on the Forbidden and Limited lists, the chance would be about 1 in 2,708.
15/40 * 12/39 * 9/38 * 6/37 * 3/36
- Outside of the initial draw, this ZTK can also occur if the opponent activates effects that allow you to draw cards during their first turn, such as "Cup of Ace". The chance of this is about 1 in 131,602 after one draw, about 1 in 32,900 after two draws, about 1 in 11,964 after three draws, about 1 in 5,264 after four draws, and about 1 in 2,622 after five draws.
- assuming no pieces of Exodia which were already drawn are discarded, the probability of drawing all remaining pieces of Exodia after
nextra turns can be calculated with the formula
(binomial(n + 5, 5) - 1) * 1/658008, where
binomial(n, k)is the binomial coefficient (often read as "n choose k"), and represents the number of ways you can draw the five pieces of Exodia (
k) in your first
ndraws; 1 is subtracted from the binomial to represent the case where all five pieces of Exodia were drawn in the first five cards (since that represents an instant win before the opponent is able to start their turn); and
1/658008is the chance of any one configuration of drawn cards (which is, as it turns out, the chance that any one case works out to, regardless of how many extra draws occur - e.g. the chance of drawing four pieces as the first four cards, and the fifth piece as the sixth card, is
5/40 * 4/39 * 3/38 * 2/37 * 35/36 * 1/35, or
4200/2763633600, which reduces to
1/658008)— Zero Turn Kill, with hidden comments shown and formatted for readability
Imagine you have a Deck of ten cards, lined up in front of you. You can number them off, left to right, from one to ten; we'll say that position one is the top of the Deck and position ten is the bottom. The card you're interested in is solid black; the rest are all solid white. Assuming they're shuffled randomly, the wanted card could be in any of the ten positions:
So if you pick a position, there's a one in ten, or 1/10, chance your card will be in that position after a random shuffle. Since the left-most card—the top card—is just another position, that means there's a 1/10 chance of drawing a card and it being the wanted card.
- Hypergeometric calculator on Stat Trek