Two cards are drawn from a deck of cards. What is the probability of drawing two hearts?
Answer (2)
Calculate the probability of the first heart: 52 13 .
Calculate the probability of the second heart given the first was a heart: 51 12 .
Multiply these probabilities: 52 13 × 51 12 = 17 1 ≈ 0.059 .
Convert to percentage: 5.9% .
Explanation
Understand the problem and provided data We want to find the probability of drawing two hearts from a standard deck of 52 cards. A standard deck has 13 hearts.
Calculate the probability of drawing the first heart The probability of drawing the first heart is the number of hearts divided by the total number of cards: P ( H 1 ) = 52 13 After drawing one heart, there are now 12 hearts left and a total of 51 cards remaining in the deck.
Calculate the probability of drawing the second heart given the first was a heart The probability of drawing a second heart, given that the first card was a heart, is: P ( H 2 ∣ H 1 ) = 51 12
Calculate the probability of drawing two hearts The probability of drawing two hearts is the product of the probability of drawing the first heart and the probability of drawing the second heart, given that the first card was a heart: P ( H 1 ∩ H 2 ) = P ( H 1 ) ⋅ P ( H 2 ∣ H 1 ) = 52 13 ⋅ 51 12 P ( H 1 ∩ H 2 ) = 52 13 ⋅ 51 12 = 2652 156 = 17 1 P ( H 1 ∩ H 2 ) ≈ 0.0588
Convert to percentage Converting this to a percentage, we get approximately 5.9%.
Examples
This type of probability calculation is used in card games like poker or bridge to determine the likelihood of getting certain hands. For example, calculating the probability of being dealt a flush (five cards of the same suit) involves similar conditional probability calculations.
The probability of drawing two hearts from a standard deck of cards is approximately 5.9%. This is calculated by finding the probabilities of drawing the first heart and the second heart sequentially. The steps demonstrate the math behind these probabilities clearly.
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