If two cards are chosen at random from a standard deck of cards, one at a time, and replaced after each pick, what is the probability that a black card is chosen first and a heart is chosen second?
Answer (1)
The probability of drawing a black card first is 52 26 = 2 1 .
The probability of drawing a heart card second is 52 13 = 4 1 .
Multiply the probabilities since the events are independent: 2 1 × 4 1 .
The probability of drawing a black card first and a heart card second is 8 1 .
Explanation
Analyze the problem and data A standard deck of cards has 52 cards, divided into four suits: spades, clubs, hearts, and diamonds. Each suit has 13 cards. Spades and clubs are black, while hearts and diamonds are red. We are drawing two cards, one at a time, with replacement. We want to find the probability of drawing a black card first and a heart second.
Calculate the probability of drawing a black card first Let's calculate the probability of drawing a black card first. There are 13 spades and 13 clubs, so there are 26 black cards in the deck. The probability of drawing a black card first is the number of black cards divided by the total number of cards: P ( " b l a c k f i rs t " ) = 52 26 = 2 1
Calculate the probability of drawing a heart card second Now, let's calculate the probability of drawing a heart second. Since we replace the first card, the deck is back to its original state of 52 cards. There are 13 hearts in the deck, so the probability of drawing a heart second is the number of hearts divided by the total number of cards: P ( " h e a r t seco n d " ) = 52 13 = 4 1
Multiply the probabilities of the two independent events Since the two events are independent (because we replace the first card), we can find the probability of both events occurring by multiplying their individual probabilities: P ( " b l a c k f i rs t an d h e a r t seco n d " ) = P ( " b l a c k f i rs t " ) × P ( " h e a r t seco n d " ) = 2 1 × 4 1 = 8 1
State the final answer The probability of drawing a black card first and a heart card second, with replacement, is 8 1 .
Examples
This type of probability problem is useful in scenarios like card games or any situation where you're drawing items from a set with replacement. For instance, imagine you're running a raffle where you draw a ticket, note the name, and then put the ticket back in before the next draw. Calculating probabilities like this helps determine the fairness and odds of winning in such games.
