##### Scenario A probability puzzle during a first-round technical screen. ##### Question You draw three cards without replacement from a standard 52-card deck. Given that among the first two cards there is at least one Ace, what is the probability that the third card is an Ace? ##### Hints Condition on the Ace count in the first two draws and apply Bayes.

Evaluates conditional probability with cards drawn without replacement after observing at least one Ace. Strong answers define the conditioning event, account for the number of Aces among the first two cards, and compute the probability that the third card is an Ace as 49/825.

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Calculate Probability of Third Card Being an Ace

Calculate the Probability of the Third Card Being an Ace

You draw three cards without replacement from a standard 52-card deck with 4 Aces and 48 non-Aces. You are told that among the first two cards there is at least one Ace.

Constraints & Assumptions

Cards are drawn without replacement.
The given information applies only to the first two cards.
Use conditional probability; do not assume the third card remains exactly 4/52 .
Give an exact fraction and an approximate decimal if useful.

Clarifying Questions to Ask

Does "among the first two" mean at least one Ace in the first two ordered draws?
Are we conditioning only on that event, or did someone intentionally reveal an Ace?
Is the deck standard and well shuffled?

Part 1 - Define Events

What events would you define to solve the conditional probability?

What This Part Should Cover

Event A : at least one Ace among the first two cards.
Event T : the third card is an Ace.
Target probability P(T | A) .

Part 2 - Compute the Probability

What is the probability that the third card is an Ace given the first two contain at least one Ace?

What This Part Should Cover

Conditioning on whether the first two contain exactly one Ace or two Aces, or using Bayes' rule directly.
Correct accounting for the remaining deck.
Exact final probability 49/825 .

What a Strong Answer Covers

A strong answer conditions on the observed information, handles without-replacement dependence correctly, and avoids the common mistake of treating the third draw as unconditioned.

Follow-up Questions

How would the answer change if exactly one of the first two cards was an Ace?
What if at least one Ace is revealed from all three cards rather than only the first two?
How would you solve it by enumerating ordered card types?

Calculate the Probability of the Third Card Being an Ace

You draw three cards without replacement from a standard 52-card deck with 4 Aces and 48 non-Aces. You are told that among the first two cards there is at least one Ace.

Constraints & Assumptions

Cards are drawn without replacement.
The given information applies only to the first two cards.
Use conditional probability; do not assume the third card remains exactly 4/52 .
Give an exact fraction and an approximate decimal if useful.

Clarifying Questions to Ask

Does "among the first two" mean at least one Ace in the first two ordered draws?
Are we conditioning only on that event, or did someone intentionally reveal an Ace?
Is the deck standard and well shuffled?

Part 1 - Define Events

What events would you define to solve the conditional probability?

What This Part Should Cover

Event A : at least one Ace among the first two cards.
Event T : the third card is an Ace.
Target probability P(T | A) .

Part 2 - Compute the Probability

What is the probability that the third card is an Ace given the first two contain at least one Ace?

What This Part Should Cover

Conditioning on whether the first two contain exactly one Ace or two Aces, or using Bayes' rule directly.
Correct accounting for the remaining deck.
Exact final probability 49/825 .

What a Strong Answer Covers

A strong answer conditions on the observed information, handles without-replacement dependence correctly, and avoids the common mistake of treating the third draw as unconditioned.

Follow-up Questions

How would the answer change if exactly one of the first two cards was an Ace?
What if at least one Ace is revealed from all three cards rather than only the first two?
How would you solve it by enumerating ordered card types?

Calculate Probability of Third Card Being an Ace

Quick Overview

Calculate Probability of Third Card Being an Ace

Calculate the Probability of the Third Card Being an Ace

Constraints & Assumptions

Clarifying Questions to Ask

Part 1 - Define Events

What This Part Should Cover

Part 2 - Compute the Probability

What This Part Should Cover

What a Strong Answer Covers

Follow-up Questions

Write your answer

Calculate Probability of Third Card Being an Ace

Quick Overview

Calculate Probability of Third Card Being an Ace

Calculate the Probability of the Third Card Being an Ace

Constraints & Assumptions

Clarifying Questions to Ask

Part 1 - Define Events

What This Part Should Cover

Part 2 - Compute the Probability

What This Part Should Cover

What a Strong Answer Covers

Follow-up Questions

Write your answer