In a standard deck of cards there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and the clubs are black and the hearts and the diamonds are red.
If two cards are chosen at random from a deck, 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?
$\frac{1}{8}$
$\frac{1}{2}$
$\frac{2}{3}$
$\frac{3}{4}$
Answer (2)
Calculate the probability of drawing a black card: 52 26 = 2 1 .
Calculate the probability of drawing a heart: 52 13 = 4 1 .
Multiply the probabilities since the events are independent: 2 1 × 4 1 = 8 1 .
The probability of drawing a black card first and a heart second is 8 1 .
Explanation
Understand the problem and provided data We are given a standard deck of 52 cards, with 13 cards in each of the four suits: spades, clubs, hearts, and diamonds. Spades and clubs are black, while hearts and diamonds are red. We want to find the probability of drawing a black card first and a heart second, with replacement.
Calculate the probability of drawing a black card First, we need to calculate the probability of drawing a black card. There are 13 spades and 13 clubs, so there are 26 black cards in total. The probability of drawing a black card is the number of black cards divided by the total number of cards: P ( Bl a c k ) = 52 26 = 2 1
Calculate the probability of drawing a heart Next, we need to calculate the probability of drawing a heart. There are 13 hearts in the deck. The probability of drawing a heart is the number of hearts divided by the total number of cards: P ( He a r t ) = 52 13 = 4 1
Calculate the combined probability Since the draws are independent (because the card is replaced after each draw), we can find the probability of drawing a black card first and a heart second by multiplying the individual probabilities: P ( Bl a c k then He a r t ) = P ( Bl a c k ) × P ( He a r t ) = 2 1 × 4 1 = 8 1
State the final answer Therefore, the probability of drawing a black card first and a heart second with replacement is 8 1 .
Examples
Consider a simplified card game where you need to draw a specific sequence of colors to win. This problem illustrates how to calculate the probability of drawing a black card followed by a heart, which is a fundamental concept in probability theory and can be applied to various scenarios involving sequential events.
The probability of drawing a black card first and a heart second from a standard deck of cards is 8 1 . This is calculated by finding the probabilities of each event and multiplying them, as the events are independent. Therefore, the correct answer is 8 1 .
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