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I play a Blue Eyes deck (handtraps count is still not complete) of 40 cards. Some cards have one copy, some have two, and some have three.

One of the most important piece of my combos is Blue Eyes Abyss Dragon, which I have one copy of. Lately, I have the Blue Eyes Abyss Dragon in my starting hand 7 games out of 10, being a brick in my hands if I do not also have Dictator of D., which I have two copies of.

How can I calculate the odds that I'll start with both these cards in my hand? More generally, how can I figure out the probability of starting with any given combo pieces?

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You can find a deck probability calculator at YuGiOh.Party (found by doing a quick web search for "YuGiOh Starting Hand")

As for how it works:

Your opening hand is 5 cards. If you have a deck of 40 cards, that means that you have 5 chances out of 40 to draw a specific card in your opening hand.

5 in 40 is the same as 1 in 8 (divide both sides by 5), or 12.5% (because 1÷8=0.125)

The odds of having 2 specific cards in your hands are then this value, multiplied by the odds of one of the other 4 cards being the other card. So, you have a 5-in-40 chance, times a 4-in-39 chance. That's 1-in-8 times 4-in-39. 1*4=4, and 8*39=312, so that's a 4 in 312 chance, or 1 in 78, which works out at 1.28%

Adding a third card means another 3 in 38 chance. 1/78 * 3/38 is 3/2964, i.e. 1 in 988, or a 0.1% chance of 3 specific cards being in your opening hand.

Of course, duplicate cards make things slightly more complicated. If you wanted a starting hand of Blue Eyes Abyss Dragon and at least 1 copy of The Melody of Awakening Dragon, then the probability would be higher than for a starting hand of Blue Eyes Abyss Dragon and Bingo Machine, Go!!!: 2.46% instead of 1.28%

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  • I guess this is the best solution we have for now I just wonder if the game uses that same probability
    – prout
    Commented Mar 11, 2023 at 15:05
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    Assuming that the game uses fair probability (why wouldn't it?), this is the mathematical answer to your question -- it's the only possible way to do it. This is also how it'd work in real life, assuming you shuffled well enough to truly randomize your deck order.
    – Schism
    Commented Mar 11, 2023 at 20:15

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