Question

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P(X = 2) equals :

• A. 52/169
• B. 25/169
• C. 49/169
• D. 24/169

Answer 1

The Correct Option is B

To solve this problem, we need to calculate the probabilities \(P(X = 1)\) and \(P(X = 2)\) when drawing two cards successively with replacement from a standard deck of 52 cards. Here, \(X\) is the random variable representing the number of aces drawn.

1. Understanding the Problem:
   • A standard deck has 4 aces and 52 cards in total.
   • We are drawing cards with replacement, meaning the deck is complete before each draw.
2. Calculating \(P(X = 1)\):
   • We can get exactly one ace by two cases: the first card is an ace, and the second card is not, or the first card is not an ace, and the second card is an ace.
   • Probability of drawing an ace (A) in one draw: \(\frac{4}{52} = \frac{1}{13}\)
   • Probability of not drawing an ace (not A) in one draw: \(\frac{48}{52} = \frac{12}{13}\)
   • Therefore, \(P(X = 1)\) is given by:
     • First card is an ace, and the second is not: \(\frac{1}{13} \times \frac{12}{13}\)
     • First card is not an ace, and the second is an ace: \(\frac{12}{13} \times \frac{1}{13}\)
     • So, \(P(X = 1) = \frac{1}{13} \times \frac{12}{13} + \frac{12}{13} \times \frac{1}{13} = \frac{24}{169}\)
3. Calculating \(P(X = 2)\):
   • This is the probability of drawing two aces: \(\frac{1}{13} \times \frac{1}{13} = \frac{1}{169}\)
4. Total Probability:
   • \(P(X = 1) + P(X = 2) = \frac{24}{169} + \frac{1}{169} = \frac{25}{169}\)
5. Conclusion: The probability that the number of aces obtained is either 1 or 2 equals \(\frac{25}{169}\).

Hence, the correct answer is \(\frac{25}{169}\).