Two cards are randomly selected from a standard 52-card deck. What is the probability of getting 2 hearts or 2 numbers less than 4 (count aces as 1)
The Correct Answer and Explanation is :
To find the probability of getting either two hearts or two cards with numbers less than 4 (where aces count as 1), we will use the principle of inclusion-exclusion.
Total Possible Outcomes
The total number of ways to choose 2 cards from a standard 52-card deck is given by the combination formula:
[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
]
For our case, ( n = 52 ) and ( k = 2 ):
[
\text{Total outcomes} = \binom{52}{2} = \frac{52 \times 51}{2} = 1326
]
Probability of Getting 2 Hearts
There are 13 hearts in a deck. The number of ways to choose 2 hearts is:
[
\text{Ways to choose 2 hearts} = \binom{13}{2} = \frac{13 \times 12}{2} = 78
]
Probability of Getting 2 Cards with Numbers Less than 4
The cards with numbers less than 4 are the aces, 2s, and 3s, totaling 12 cards (4 of each suit). The number of ways to choose 2 cards from these is:
[
\text{Ways to choose 2 cards < 4} = \binom{12}{2} = \frac{12 \times 11}{2} = 66
]
Probability of Getting Both Events (2 Hearts and 2 Cards < 4)
Since the events “getting 2 hearts” and “getting 2 cards with numbers less than 4” are mutually exclusive (you cannot have 2 hearts that are also < 4, as they are not less than 4), we simply add the probabilities of both events.
Combined Probability
The total number of favorable outcomes for either event is:
[
\text{Favorable outcomes} = \text{Ways to choose 2 hearts} + \text{Ways to choose 2 cards < 4} = 78 + 66 = 144
]
Probability Calculation
Finally, the probability of getting either 2 hearts or 2 cards with numbers less than 4 is given by the formula:
[
P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{144}{1326} \approx 0.1087
]
Conclusion
Thus, the probability of drawing either 2 hearts or 2 cards with values less than 4 is approximately 0.1087 or 10.87%.