Brain Teasers · Drawing for the First Ace
Expected Cards to the First Ace - Play It (53/5)
Turn a shuffled deck over one card at a time. On average, how many cards until the first ace?
How do you find the expected position of the first ace?
Linearity of expectation lets you add up, card by card, the chance each ordinary card comes before every ace, even though those events depend on each other.
By symmetry, each of the 48 non-aces is equally likely to fall in any of the five gaps the aces create.
How it works
A shuffled 52-card deck, turned over one card at a time until the first ace. The story asks how many cards you turn on average, counting the ace. You type your answer and check it. Then you step through the answer and the breakdown.
How scoring works
This teaser checks your answer. Any correct form counts - a fraction, a decimal rounded correctly, a percent, or the number in words. You get two tries before the answer opens. A correct first try earns a ticket; a second try or opening the answer early is recorded but earns nothing. The story teasers share one lifetime unlock for free users.
The key insight
Each non-ace comes before all four aces with probability 1/5, so on average 48/5 = 9.6 cards come first. Add the ace: 53/5 = 10.6.
The full step-by-step solution lives at /brain-teasers/first-ace/solution.
A worked example
In general the first of k special cards in N is at position (N + 1)/(k + 1) on average. With 3 cards and 1 ace, that is 2, which matches 1st, 2nd and 3rd being equally likely.
Common mistakes
• Answering 13 by dividing 52 by 4.
• Giving 9.6 when the question counts the ace itself.
• Writing out a 49-term sum instead of using symmetry.
Why interviews test this
Interviewers want the gaps argument and the linearity step, not a long sum. The written solution is at /brain-teasers/first-ace/solution.
Play Drawing for the First Ace on Outcry
Drawing for the First Ace is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play Drawing for the First Ace and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.