Problem 20 - Algebra and Trigonometry

Question

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a card greater than 3 and less than 7.}$$

Step-by-Step Solution

Answer

The probability of drawing a card greater than 3 and less than 7 from a standard 52-card deck is \(\frac{3}{13}\).

Step 1: Identify the sample space and success event

The sample space, denoted by S, of a random experiment is the set of all possible outcomes. In this case, it's every card in the deck, so there are 52 possible outcomes. The success event, which we'll denote by E, is drawing a card greater than 3 and less than 7. Thus, the cards fitting this requirement are 4, 5, and 6. Since each of these cards comes in 4 different suits (hearts, diamonds, clubs, spades), there are 12 successful outcomes.

Step 2: Apply the formula for probability

The probability of an event is given by the ratio of the number of successful outcomes to the number of possible outcomes. Using the notation established, it can be denoted as: \[ P(E) = \frac{n(E)}{n(S)} \] where \(n(E)\) is the number of successful outcomes and \(n(S)\) is the number of possible outcomes.

Step 3: Calculate the Probability

Substitute the values into the formula: \[ P(E) = \frac{n(E)}{n(S)} = \frac{12}{52} = \frac{3}{13} \].

Key Concepts

Sample Space

In the realm of probability, the sample space is the complete set of all possible outcomes of a random experiment. When drawing a card from a standard 52-card deck, the sample space includes every card in the deck. This means that there are 52 different outcomes, one for each card. For any specific probability question involving this deck, knowing the sample space is essential, as it forms the denominator in the probability equation. In this exercise, the deck comprises 52 cards, thus our sample space ( (S) ) is 52.

Successful Outcomes

Successful outcomes refer to those specific outcomes that satisfy the conditions of an event. In this exercise, we are interested in cards greater than 3 and less than 7. This condition isolates the cards numbered 4, 5, and 6. Each of these numbers is represented in all four suits: hearts, diamonds, clubs, and spades. Therefore, there are 3 card values, multiplied by 4 suits, totaling 12 successful outcomes. Identifying successful outcomes precisely is crucial because they determine the numerator in the probability calculation.

Standard 52-Card Deck

A standard 52-card deck is a common tool in probability exercises and card games. It consists of four suits: hearts, diamonds, clubs, and spades. Each suit has 13 ranks starting from Ace through to King. These ranks typically include numerical cards (2 through 10) and face cards (Jack, Queen, King, and sometimes Ace, depending on the context). Understanding this structure is important when solving problems, as it helps in visualizing the sample space and successful outcomes. The properties of the deck ensure the sample space is uniform, meaning each card has an equal probability of being drawn.

Probability Formula

The probability of an event is calculated using the probability formula, which considers both successful outcomes and the sample space. The formula is expressed as:

• \( P(E) = \frac{n(E)}{n(S)} \)
• \( n(E) \) is the number of successful outcomes.
• \( n(S) \) is the total number of possible outcomes in the sample space.

In this scenario, the successful outcomes are the 12 ways to draw a card that is greater than 3 and less than 7, and the sample space is the 52 cards of the deck. Therefore, the probability is:

• \( P(E) = \frac{12}{52} = \frac{3}{13} \)

This formula provides a straightforward method to determine the likelihood of any given event, as long as you correctly count both the successful outcomes and the total possibilities.