Final answer:
The probability that a randomly dealt 3-card hand from a standard deck of 52 cards contains all red cards is approximately 11.76%.
Step-by-step explanation:
The question relates to the calculation of the probability that a randomly dealt 3-card hand from a standard deck of 52 cards contains all red cards. To calculate this, we must first note that there are two red suits in the deck: hearts and diamonds, each with 13 cards, making a total of 26 red cards.
To find the probability, we use the combination formula to determine the number of ways to select 3 red cards out of the 26 available, and divide this by the total number of ways to select any 3 cards from the entire deck of 52 cards.
The formula for a combination is C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, k is the number of items to choose, and ! signifies a factorial.
- Calculate the number of ways to choose 3 red cards from the 26 red cards: C(26, 3).
- Calculate the total number of ways to choose 3 cards from the deck: C(52, 3).
- Divide the number of red combinations by the total combinations to get the probability.
Performing these calculations:
C(26, 3) = 26! / (3! * (26-3)!) = 2,600
C(52, 3) = 52! / (3! * (49)!) = 22,100
Probability = 2,600 / 22,100 = 0.1176 or about 11.76% chance.