Introduction

There are 22 Tarot cards in Ogre Battle - the Major Arcana. Starting a new game brings up the iconic scene of being dealt 6 Tarot cards by Warren from the deck and then drawing a 7th bonus card. Can visibly count 16 cards from the deck at the bonus pull, implying duplicate cards in the starting hand are impossible.

With the assumption of perfect RNG, as in, each hand combination is equally likely and one card being delt doesn't bias the odds of the remaining cards, probabilities can be exactly calculated.

That said, the RNG can be manipulated, the starting seed has weak entropy and I've pulled two Worlds multiple times on PSX. The following odds give a good baseline of how much resetting you're apt to do per viable run with no RNG manipulation, else the relative difficulty in finding a starting seed with the cards you want.

I don't think most people care to see the calculations using combinatorics and the hypergeometric distribution. Just gimme the numbers!

• Total combinations of 6 cards where order doesn't matter: 74613
• Total combinations of 7 cards where order doesn't matter: 170544

These are rookie numbers in Computer Science, allowing generation of all deck combinations in under 1 second that verify the below calculations. They become the denominators with numerators the number of desired combinations.

Odds in fraction form are exact.

Odds

• Desired Tarot pulled in the first 6 cards: 20349 / 74613
  = 27.27%, 27 repeating of course Example: Fool

All further odds use all 7 cards

• Specific card pulled (in 7 cards): 54264 / 170544
  = 31.81%, 81 repeating of course
  = 1 in 3.67 attempts
  Example: Fool

• At least 1 of 2 specific cards pulled: 93024 / 170544
  = 54.54%, 54 repeating of course
  = 1 in 1.83 attempts
  Example: Magician and/or Justice

• Specific card AND (at least 1 of 2 other specific cards): 27132 / 170544
  = 15.90%, 90 repeating of course
  = 1 in 6.29 attempts
  Example: Fool AND Magician, Fool AND Justice, Fool AND Magician AND Justice

• Specific two cards: 15504 / 170544
  = 9.09%, 09 repeating of course: 6x less than odds of at least 1 of 2
  = 1 in 11 attempts, exactly
  Example: Fool AND Hermit

• At least 1 of 3 specific cards pulled: 120156 / 170544
  = 70.45%, 45 repeating of course
  = 1 in 1.42 attempts
  Example: Magician, OR Justice, OR Judgment, OR 2 out of 3, OR all 3

• At least two out of three specific cards: 38760 / 170544
  = 22.27%, 27 repeating of course
  = 1 in 4.4 attempts, exactly
  Example: Magician AND Justice, OR Magician AND Judgment, OR Justice AND Judgment, OR all 3

• Specific card AND (at least 1 of 3 other specific cards): 35700 / 170544
  = 20.93% rounded
  = 1 in 4.78 attempts
  Example: Fool combined with the possibilities of the previous at least 1 of 3 other cards

• Specific card AND at least two out of three other specific cards: 9996 / 170544
  = 5.86% rounded
  = 1 in 17.06 attempts
  Example: Fool combined with the possibilities of the previous at least 2 of 3 other cards

• Specific three cards: 3876 / 170544
  = 2.27%, 27 repeating of course
  = 1 in 44 attempts, exactly
  Example: Magician AND Justice AND Judgment

• Specific 4 cards: 816 / 170544
  = 0.4785% rounded
  = 1 in 209 attempts, exactly
  Example: Fool AND Magician AND Justice AND Judgment

Kind of interesting that the probability of at least 1 out of 3 specific cards is 31x greater than the odds of 3 specific cards. The probability of at least 2 out of 3 specific cards is 10x greater than 3 specific cards. Probably of at least 1 of 2 specific cards is 6x greater than 2 specific cards. Probability of 2 specific cards is 19x greater than 4 specific cards.

Requests

I'm happy to calculate the Tarot card odds of any combination you want, else iterate through all hands via software and count matches if the combination is particularly complex.

Credit

You may freely distribute and share these odds anywhere except on a wiki that has advertisements or a website related to the for-profit wiki owners. No credit needed or expected. Other for-profit website or message board not related to the wiki owners, totally fine.

Credit requested if you post the combinatorics or hypergeometric calculations or use my (very hard to use) software. Same restrictions apply.