Problem 39 - Algebra and Trigonometry

Question

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a king.

Step-by-Step Solution

Answer

The probability of not being dealt a King is \(\frac{48}{52} = \frac{12}{13}\).

Step 1: Identify Total Outcomes

When a single card is drawn from a standard deck of 52 cards, there are 52 possible outcomes because each card represents a different outcome.

Step 2: Identify Favorable Outcomes

The question asks for the chances of not drawing a King. In a standard deck of 52 cards, there are 4 Kings. Thus, the favorable outcomes for not drawing a King is \(52 - 4 = 48\).

Step 3: Calculate Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes. Therefore, the probability of not drawing a King is \(\frac{48}{52}\).

Key Concepts

52-card deck

A standard 52-card deck is like a universe of its own where the art of probability comes alive. In this deck, you'll find four suits: hearts, diamonds, clubs, and spades. Each suit holds 13 cards, amounting to the total of 52 cards.
The deck includes:

• 13 hearts
• 13 diamonds
• 13 clubs
• 13 spades

Each suit has one King, making a total of four Kings in the deck. This uniform distribution of cards forms the basis for various probability experiments. Whenever we draw a card, we pick from these 52 cards, leading to 52 possible outcomes for each draw.

favorable outcomes

Favorable outcomes in probability refer to the specific outcomes that we are interested in. For this problem, these are the outcomes where a King is not drawn from the deck.
Out of the 52 cards, 4 are Kings, leaving us with 48 cards that are not Kings. These 48 are our favorable outcomes when the event is 'drawing a card that is not a King'.
By focusing on favorable outcomes, we can determine how likely it is for our event of interest to occur in the context of all possible outcomes.

total outcomes

Total outcomes describe all the potential results that can occur in a probability scenario. For a single card drawn from our 52-card deck, the total outcomes are straightforward — there are 52 possible cards (or outcomes) that could be drawn.
These include every heart, diamond, club, and spade from Ace through to King. Having a clear idea of the total number of outcomes helps us establish the denominator in probability calculations. Here, the 52 outcomes form the set of possibilities from which favorable outcomes are selected.

probability calculation

Probability calculation helps in quantifying the uncertainty of events. It measures how likely an event is to happen by comparing the number of favorable outcomes to the total number of outcomes.
The formula for probability is given by:\[P( ext{Event}) = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Number of Outcomes}}\]In the context of our problem, the probability of not drawing a King is determined as:

• Number of favorable outcomes = 48 (cards that are not Kings)
• Total outcomes = 52 (all cards in the deck)

Applying the formula:\[P( ext{Not drawing a King}) = \frac{48}{52} = \frac{12}{13}\]Thus, the chance of drawing a card that is not a King is approximately 92.3% when expressed as a percentage.