Non-Ace as Second Card

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

2. First Card is not an Ace:

   • Probability of drawing a non-Ace first: $\frac{48}{52}$
   • 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: $\frac{47}{51}$
   • Combined probability for this scenario: $P(\text{1st not Ace, 2nd not Ace}) = \frac{48}{52} \times \frac{47}{51}$

Total Probability Calculation

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

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

Substituting the values:

$P(\text{2nd not Ace}) = \left(\frac{4}{52} \times \frac{48}{51}\right) + \left(\frac{48}{52} \times \frac{47}{51}\right)$

Calculating each term:

• $\frac{4}{52} \times \frac{48}{51} = \frac{192}{2652} \approx 0.07239$
• $\frac{48}{52} \times \frac{47}{51} = \frac{2256}{2652} \approx 0.85067$

Adding these probabilities:

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

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.