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Chapter 7: Problem 24

Two cards are drawn without replacement from a wellshuffled deck of 52 playing cards. a. What is the probability that the first card drawn is a heart? b. What is the probability that the second card drawn is a heart if the first card drawn was not a heart? c. What is the probability that the second card drawn is a heart if the first card drawn was a heart?

Short Answer

Expert verified
a) The probability that the first card drawn is a heart is \(\frac{1}{4}\). b) The probability that the second card drawn is a heart, given that the first card drawn is not a heart, is \(\frac{13}{51}\). c) The probability that the second card drawn is a heart, given that the first card drawn is a heart, is \(\frac{12}{51}\).

Step by step solution

01

In a deck of 52 playing cards, there are four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards. We are supposed to find the probability of drawing a heart in three different scenarios. The given deck is well-shuffled, and the cards are drawn without replacement. #Step 2: Calculating the probability of the first card being a heart#

There are 13 hearts in the deck, and we have to find the probability that the first card drawn is a heart. The total number of cards is 52. So, the probability of the first card being a heart is: \(P(1st \: card \: is \: a \: heart) = \frac{Number \: of \: hearts}{Total \: number \: of \: cards} = \frac{13}{52}=\frac{1}{4}\) The probability that the first card drawn is a heart is \(\frac{1}{4}\). #Step 3: Calculating the probability of the second card being a heart, given that the first card is not a heart#
02

If the first card is not a heart, then there are still 13 hearts remaining in the deck and only 51 cards left in the deck. So, the probability of the second card being a heart in this scenario is: \(P(2nd \: card \: is \: a \: heart | 1st \: card \: is \: not \: a \: heart) = \frac{Number \: of \: remaining \: hearts}{Total \:number \:of \:remaining \:cards}=\frac{13}{51}\) The probability that the second card drawn is a heart, given that the first card drawn is not a heart, is \(\frac{13}{51}\). #Step 4: Calculating the probability of the second card being a heart, given that the first card is a heart#

If the first card is a heart, then there are 12 hearts remaining in the deck and only 51 cards left in the deck. So, the probability of the second card being a heart in this scenario is: \(P(2nd \: card \: is \: a \: heart | 1st \: card \: is \: a \: heart) = \frac{Number \: of \: remaining \: hearts}{Total \:number \:of \:remaining \:cards}=\frac{12}{51}\) The probability that the second card drawn is a heart, given that the first card drawn is a heart, is \(\frac{12}{51}\). So, the answers are as follows: a) The probability that the first card drawn is a heart is \(\boxed{\frac{1}{4}}\). b) The probability that the second card drawn is a heart, given that the first card drawn is not a heart, is \(\boxed{\frac{13}{51}}\). c) The probability that the second card drawn is a heart, given that the first card drawn is a heart, is \(\boxed{\frac{12}{51}}\).

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