A standard deck of cards has 52 cards, 13 of each of the 4 suits: hearts, diamonds, clubs, and spades. After the deck, Linbo deals the cards to his friends for the game. Which expression can be used to find the probability the first three cards that he deals are all hearts if none of them are replaced?
$\left(\frac{13}{52}\right)\left(\frac{12}{52}\right)\left(\frac{11}{52}\right)$
$\left(\frac{13}{52}\right)\left(\frac{12}{51}\right)\left(\frac{11}{50}\right)$
$\left(\frac{13}{52}\right)\left(\frac{13}{52}\right)\left(\frac{13}{52}\right)$
$\left(\frac{13}{52}\right)\left(\frac{13}{51}\right)\left(\frac{13}{50}\right)$
Answer (1)
The probability of the first card being a heart is 52 13 .
The probability of the second card being a heart, given the first was a heart, is 51 12 .
The probability of the third card being a heart, given the first two were hearts, is 50 11 .
The overall probability is ( 52 13 ) ( 51 12 ) ( 50 11 ) .
Explanation
Understand the problem We are given a problem about finding the probability of drawing three hearts in a row from a standard deck of 52 cards without replacement. This means that once a card is drawn, it is not put back into the deck.
Calculate the probability of the first card being a heart The probability of drawing a heart on the first draw is the number of hearts divided by the total number of cards. Since there are 13 hearts in a deck of 52 cards, the probability of drawing a heart on the first draw is 52 13 .
Calculate the probability of the second card being a heart After drawing a heart on the first draw, there are now only 12 hearts left in the deck, and the total number of cards is now 51. Therefore, the probability of drawing a heart on the second draw, given that a heart was drawn on the first draw, is 51 12 .
Calculate the probability of the third card being a heart After drawing hearts on the first two draws, there are now only 11 hearts left in the deck, and the total number of cards is now 50. Therefore, the probability of drawing a heart on the third draw, given that hearts were drawn on the first two draws, is 50 11 .
Calculate the overall probability To find the probability of all three events happening in sequence, we multiply the probabilities of each event: 52 13 × 51 12 × 50 11 . This corresponds to the expression ( 52 13 ) ( 51 12 ) ( 50 11 ) .
State the final answer The expression that can be used to find the probability that the first three cards dealt are all hearts if none of them are replaced is ( 52 13 ) ( 51 12 ) ( 50 11 ) .
Examples
This type of probability calculation is used in many card games to determine the likelihood of getting certain hands. For example, in poker, players often calculate the probability of getting a flush (five cards of the same suit) or a straight (five cards in sequence). Understanding these probabilities helps players make informed decisions about betting and playing their hands. Similarly, in other games, calculating the probability of drawing specific cards can help players strategize and increase their chances of winning.
