In an ideal random shuffle, you start Klondike with an Ace showing about 45% of the time. That is a little less than half of all deals. Starting without a visible Ace is normal, and it does not tell you that the game is lost. The exact chance of at least one opening Ace is 44.9644%.
Method: exact mathematical calculation, not an experiment or a sample of player games. We use a 52-card deck with four Aces and no Jokers. Every order of the deck is equally likely. We count the seven face-up cards before any card is moved or drawn. These results are not a measurement of this website’s seeded deal generator or its selected deal pools.
Why we count only the seven face-up cards
The Klondike setup puts 28 cards in the tableau, with seven face up and 21 face down. With every deck order equally likely, any seven fixed positions have the same chance of containing Aces. The exposed cards do not need to be the first seven cards dealt for the calculation to work.
Write C(n, k) for the number of ways to choose k objects from n without regard to order. There are C(52, 7) possible sets of exposed cards. A set with no Ace must select all seven cards from the 48 non-Aces.
P(at least one Ace) = 1 − C(48, 7) / C(52, 7)
This is an application of the hypergeometric distribution, which describes sampling without replacement. Treating the seven cards as independent draws with replacement would give a different answer.
How often each opening happens
| Aces showing | Probability |
|---|---|
| 0 | 55.0356% |
| 1 | 36.6904% |
| 2 | 7.6794% |
| 3 | 0.5818% |
| 4 | 0.0129% |
For exactly k Aces, choose k of the four Aces and 7 − k of the non-Aces, then divide by all possible sets: C(4, k) × C(48, 7 − k) / C(52, 7). The exact probabilities sum to one; displayed rounding can affect a hand-added total.
What changes after you see the opening?
If none of the seven face-up cards is an Ace, all four Aces must be among the 45 unseen cards. Based only on that information, a particular hidden position has a 4 in 45 chance of holding an Ace: about 8.9%. The chance changes because you now know where seven non-Aces are.
This does not make every possible move equally valuable. Some moves reveal a hidden tableau card; others merely rearrange visible cards. These odds help describe the unseen cards. They do not tell you which move is best or promise that the next card will be an Ace.
Check the calculation yourself
The code below is for readers who want to check the arithmetic. You do not need it to use the main result: an opening with no Ace showing is slightly more common than one with an Ace.
from math import comb
for k in range(5):
p = comb(4, k) * comb(48, 7-k) / comb(52, 7)
print(k, f"{100*p:.4f}%")
print(1 - comb(48, 7) / comb(52, 7))The same calculations are included in our downloadable reproduction script. It uses Python’s standard library and does not fetch player data.
To see what we are counting, deal a game of classic Solitaire and count only the initial seven exposed tableau cards, before drawing or moving. One opening is an illustration, not a test of the model. For outcomes rather than opening ranks, use the separate win-rate report and its stated methodology.