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Data Interview Question

Non-Ace as Second Card

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Solution & Explanation

To solve the problem of finding the probability that the second card drawn from a well-shuffled deck is not an Ace, we need to consider the possible scenarios and use basic probability principles.

Understanding the Deck

  • A standard deck has 52 cards.
  • There are 4 Aces in the deck.
  • Therefore, there are 48 non-Ace cards in the deck.

Scenarios of Drawing Two Cards

  1. First Card is an Ace:

    • Probability of drawing an Ace first: 452\frac{4}{52}
    • If the first card is an Ace, there are 51 cards left, with 3 being Aces.
    • Probability of second card being not an Ace: 4851\frac{48}{51}
    • Combined probability for this scenario: P(1st Ace, 2nd not Ace)=452×4851P(\text{1st Ace, 2nd not Ace}) = \frac{4}{52} \times \frac{48}{51}
  2. First Card is not an Ace:

    • Probability of drawing a non-Ace first: 4852\frac{48}{52}
    • If the first card is not an Ace, there are still 4 Aces among the 51 cards.
    • Probability of second card being not an Ace: 4751\frac{47}{51}
    • Combined probability for this scenario: P(1st not Ace, 2nd not Ace)=4852×4751P(\text{1st not Ace, 2nd not Ace}) = \frac{48}{52} \times \frac{47}{51}

Total Probability Calculation

The total probability that the second card is not an Ace is the sum of probabilities of the two scenarios:

P(2nd not Ace)=P(1st Ace, 2nd not Ace)+P(1st not Ace, 2nd not Ace)P(\text{2nd not Ace}) = P(\text{1st Ace, 2nd not Ace}) + P(\text{1st not Ace, 2nd not Ace})

Substituting the values:

P(2nd not Ace)=(452×4851)+(4852×4751)P(\text{2nd not Ace}) = \left(\frac{4}{52} \times \frac{48}{51}\right) + \left(\frac{48}{52} \times \frac{47}{51}\right)

Calculating each term:

  • 452×4851=19226520.07239\frac{4}{52} \times \frac{48}{51} = \frac{192}{2652} \approx 0.07239
  • 4852×4751=225626520.85067\frac{48}{52} \times \frac{47}{51} = \frac{2256}{2652} \approx 0.85067

Adding these probabilities:

P(2nd not Ace)=0.07239+0.85067=0.92306P(\text{2nd not Ace}) = 0.07239 + 0.85067 = 0.92306

Conclusion

The probability that the second card drawn is not an Ace is approximately 92.31%. This solution demonstrates that despite the first card drawn, the likelihood of the second card not being an Ace remains quite high.