Question

You draw a card at random from a standard deck of 52 cards. Find each of the following conditional probabilities: a) The card is a heart, given that it is red. b) The card is red, given that it is a heart. c) The card is an ace, given that it is red. d) The card is a queen, given that it is a face card.

Short answer

a) \(\frac{1}{2}\), b) 1, c) \(\frac{1}{13}\), d) \(\frac{1}{3}\)

Step-by-step solution

Define the Problem

Understand that we are looking for various conditional probabilities from a standard deck of 52 cards. A standard deck includes 4 suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red, while clubs and spades are black. Each suit has 13 cards, including numbered cards (2 through 10) and face cards (jack, queen, king), plus an ace.

Calculate Probability for Part (a)

We need to find the probability that a card is a heart given it is red. There are 26 red cards in the deck (13 hearts and 13 diamonds). Therefore, the probability that a card is a heart given it is red is the number of hearts divided by the total number of red cards: \( P(\text{Heart} | \text{Red}) = \frac{13}{26} = \frac{1}{2} \).

Calculate Probability for Part (b)

The probability that a card is red given it is a heart is straightforward since all hearts are red. So the probability is \( P(\text{Red} | \text{Heart}) = 1 \).

Calculate Probability for Part (c)

To find the probability that a card is an ace given it is red, note there are 2 red aces in the deck (the ace of hearts and ace of diamonds). So this probability is \( P(\text{Ace} | \text{Red}) = \frac{2}{26} = \frac{1}{13} \).

Calculate Probability for Part (d)

The probability that a card is a queen given it is a face card. There are three face cards for each of the four suits (jack, queen, king), so 12 face cards total. There are 4 queens in the deck. Hence, \( P(\text{Queen} | \text{Face Card}) = \frac{4}{12} = \frac{1}{3} \).

Key concepts

Standard Deck of Cards

A standard deck of cards is a common tool used in probability exercises. It consists of 52 cards, divided equally among four suits: hearts, diamonds, clubs, and spades. Both hearts and diamonds are red suits, while clubs and spades are black suits. Each suit has 13 cards, which include numbered cards from 2 to 10, and three face cards: jack, queen, and king, plus an ace. Understanding the composition of a standard deck is crucial when calculating probabilities, especially when dealing with conditional probabilities. Knowing how the suits and cards are divided allows us to quickly assess how often certain conditions might occur in various scenarios, such as picking a card at random.

Probability Calculation

Calculating probability involves determining the likelihood of an event occurring. This is done by dividing the number of favorable outcomes by the total number of possible outcomes. For instance, if we want to calculate the probability of drawing a heart from a standard deck of cards, we note there are 13 hearts in the entire deck of 52 cards. Therefore, the probability is: \[ P( ext{Heart}) = \frac{13}{52} = \frac{1}{4} \].In exercises involving conditional probability, such as finding the chance of drawing a specific card given a particular condition, the approach is slightly adjusted, focusing only on the subset of the deck that meets the given condition. This adjustment forms the foundation for more complex probability scenarios.

Card Suit Probabilities

In the context of card games and probability exercises, understanding the likelihood of drawing a particular suit can be very useful. Each suit in a standard deck has an equal distribution:

• 13 cards in each suit
• Each suit corresponds to 25% or \( \frac{1}{4} \) of the deck

Let’s explore a specific scenario: calculating the probability that a randomly drawn card is a heart given that it is red. Since red cards consist of hearts and diamonds, we have 26 red cards in total (13 hearts and 13 diamonds). Therefore, the probability of drawing a heart given that the card is red is:\[ P( ext{Heart} | ext{Red}) = \frac{13}{26} = \frac{1}{2} \]. This means that if you already know a card is red, there’s a 50% chance it’s a heart, precisely highlighting the use of subset analysis in conditional probability.

Event Given Probability

Conditional probability focuses on the probability of an event occurring given that another event has already occurred. In our card deck scenario, this involves narrowing down the pool of potential cards to those that meet a specific condition.For example, let's consider calculating the probability that a card is an ace given that it is red. In a standard deck, there are two red aces – one from hearts and one from diamonds. This means our focus is on these specific outcomes rather than the entire deck:\[ P( ext{Ace} | ext{Red}) = \frac{2}{26} = \frac{1}{13} \].This conditional approach helps to refine our probability calculations by focusing only on relevant factors, making the analysis more precise and relevant to the given scenario.