What are the odds of drawing all five Exodia pieces in an opening hand?

From a 40-card deck, a 5-card opening hand contains all five Exodia pieces with probability 1 / C(40,5) = 1 / 658,008.

Exodia probability

Five pieces. One opening hand. Almost never.

A 5-card opening hand from a 40-card deck. To win on turn one you need all five forbidden pieces at once — a single hand out of 658,008.

The five pieces you need

Right Arm

Right Leg

Left Arm

Left Leg

Head

?—

Press Draw hand to deal five cards off a shuffled deck. The odds of lighting all five slots are 1 in 658,008.

Empirical rate

—

0 Exodia in 0 hands

Theoretical · C(40,5)

1 / 658,008

P ≈ 0.000152%

Deck size40 cards

Yu-Gi-Oh decks run 40–60. Smaller deck → fewer ways to draw 5 → better Exodia odds. The slider recomputes C(n,5) live.

Deal hands

Empirical rate vs theory0 hands

empirical 1/N theory 1/658,008

How it works

Each hand shuffles a deck of n cards — exactly 5 of which are Exodia pieces — and deals the top 5. The slots up top light gold for every piece that lands in your hand; all five lit is the jackpot.

Single draws are theatre. The truth shows up in bulk: hit Run 100,000 and watch the empirical 1/N rate plotted against the flat theoretical line. At these odds you may finish 100k hands with zero jackpots — that is the lesson.

The math

Drawing 5 specific cards from 40 (no replacement):

P = (35 choose 0) × (5 choose 5) / (40 choose 5) = 1 / 658,008

≈ 0.000152%. At 1 game per minute, ~457 days of nonstop drawing for one expected Exodia hand.

In a 60-card deck, odds drop to 1 / 5,461,512. That's why competitive Exodia decks run the 40-minimum.

Why duelists hate it

Standard Exodia decks don't wait for the hand — they use search/draw cards (Pot of Greed, Painful Choice, One Day of Peace) to fish for pieces. Real consistency is closer to 1 in 8–15 with a competitive build.

The raw 1 / 658,008 figure is for pure opening hand with zero deck manipulation. That's the version Pegasus's intro music expects.

Formula breakdown

(40 choose 5) = 40! / (5! · 35!) = 658,008 distinct 5-card opening hands.

Of those, exactly one contains the 5 Exodia pieces. So P = 1 / 658,008.

This is hypergeometric, not binomial — cards don't go back. Same math as the lottery, with smaller numbers.

Citations

Feller (1968) — Introduction to Probability Theory, Vol. I. Hypergeometric distribution.
Konami (2002+) — Yu-Gi-Oh! TCG official rulebook: 40-card minimum deck, 5-card opening hand.
Grinstead & Snell (1997) — Introduction to Probability, ch. on combinatorial probability.
Ross (2014) — A First Course in Probability, 9th ed., binomial coefficients.

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