Question

What is the probability that a card drawn at random from a 52 -card deck will be a heart or an even-numbered card?

Short answer

The probability is \( \frac{7}{13} \).

Step-by-step solution

Identify the Total Number of Cards

The standard deck of playing cards contains 52 cards. This is our total sample space for the problem.

Count the Number of Hearts

In a deck of cards, there are 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards. Therefore, there are 13 hearts in the deck.

Count the Number of Even-Numbered Cards

Even-numbered cards, in each suit, include the 2, 4, 6, 8, and 10. There are 5 even-numbered cards per suit. As there are 4 suits, this results in a total of 5 x 4 = 20 even-numbered cards in the deck.

Identify Overlapping Cards

Since we counted cards by their suit and number, some cards may have been counted twice. These are even-numbered hearts. The even-numbered cards in the hearts suit are the 2, 4, 6, 8, and 10, contributing to 5 overlapping cards.

Apply the Inclusion-Exclusion Principle

Using the formula for calculating probability of union, we have: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Here, \( P(A) \) is the probability of drawing a heart, \( P(B) \) is the probability of drawing an even card, and \( P(A \cap B) \) accounts for cards counted in both categories. So, we have:\[ P( ext{heart or even}) = \frac{13}{52} + \frac{20}{52} - \frac{5}{52} \].

Calculate the Final Probability

Substitute values in the formula and calculate: \[ P( ext{heart or even}) = \frac{13 + 20 - 5}{52} = \frac{28}{52} = \frac{7}{13} \].

Key concepts

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a fundamental concept in probability theory, used to calculate the probability of the union of two events. When we want to find the probability that one of several events occurs, and these events are not mutually exclusive, this is where the inclusion-exclusion principle becomes valuable. It helps us correct for any double-counted outcomes that happened when we considered overlapping events.

For example, when calculating the probability of drawing a heart or an even-numbered card from a standard deck, some cards (the even-numbered hearts) fit both categories. These cards would be counted twice if we simply added the number of hearts and the number of even cards. The principle tells us to subtract the overlap, ensuring that each card is only counted once. The formula is:

• The probability of A or B ( \( P(A \cup B) \) ) = Probability of A ( \( P(A) \) ) + Probability of B ( \( P(B) \) ) - Probability of both A and B ( \( P(A \cap B) \) ).

This method precisely calculates the probability without overcounting any single outcome.

Standard Deck of Cards

A standard deck of playing cards is a common fixture in probability exercises. It consists of 52 cards divided equally among four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, including number cards ranging from 2 to 10, and three face cards: Jack, Queen, and King. Additionally, there is an Ace in each suit, which can sometimes serve different values depending on the game.

In problem-solving, knowing the structure of a standard deck makes it easier to categorize and count cards based on certain criteria, such as suits or card numbers. This basic configuration provides a controlled sample space for a multitude of probability-related problems. Understanding this foundational setup is vital for applying principles like inclusion-exclusion in more complex problems.

Even-Numbered Cards

In a deck of 52 cards, defining which cards are considered even-numbered is crucial. Even-numbered cards are those whose face values are 2, 4, 6, 8, and 10. Each suit in the deck—hearts, diamonds, clubs, and spades—contains exactly these five even-numbered cards. This means there are 5 even-numbered cards per suit.

To find the total number of even-numbered cards in the deck, you would multiply the number of even cards per suit (5) by the number of suits (4), resulting in 20 even-numbered cards in the whole deck. This count is essential when applying the inclusion-exclusion principle to determine the probability of drawing either an even card or a card from another specific group, such as hearts. Recognizing these cards and their distribution helps clearly define the parameters for calculating probabilities accurately.