Question

A standard deck of cards has 52 cards, 13 of each of the 4 suits: hearts, diamonds, clubs, and spades. After shuffling the deck, Linbo deals the cards to his friends for the game. Which expression can be used to find the probability that the first three cards that he deals are all hearts if none of them are replaced?
(StartFraction 13 over 52 EndFraction) (StartFraction 12 over 52 EndFraction) (StartFraction 11 over 52 EndFraction)
(StartFraction 13 over 52 EndFraction) (StartFraction 12 over 51 EndFraction) (StartFraction 11 over 50 EndFraction)
(StartFraction 13 over 52 EndFraction) (StartFraction 13 over 52 EndFraction) (StartFraction 13 over 52 EndFraction)
(StartFraction 13 over 52 EndFraction) (StartFraction 13 over 51 EndFraction) (StartFraction 13 over 50 EndFraction)

Answer 1

To find the probability that the first three cards dealt are all hearts when none of them are replaced, we must consider the total number of hearts and the total number of cards remaining after each draw.

1. The probability that the first card dealt is a heart: There are 13 hearts in a total of 52 cards.
\[
P(\text{first card is heart}) = \frac{13}{52}
\]

2. After dealing the first heart, there are now 12 hearts left in a total of 51 cards.
\[
P(\text{second card is heart}) = \frac{12}{51}
\]

3. After dealing the second heart, there are now 11 hearts left in a total of 50 cards.
\[
P(\text{third card is heart}) = \frac{11}{50}
\]

Combining these probabilities, the expression for the probability that the first three cards dealt are all hearts is:
\[
P(\text{first three cards are hearts}) = \left(\frac{13}{52}\right) \left(\frac{12}{51}\right) \left(\frac{11}{50}\right)
\]

Thus, the correct expression is:
\[
\left(\frac{13}{52}\right) \left(\frac{12}{51}\right) \left(\frac{11}{50}\right)
\]

So the answer is:
\[
\left(\frac{13}{52}\right) \left(\frac{12}{51}\right) \left(\frac{11}{50}\right)
\]