Problem 4 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The probability of not being dealt a club is \(\frac{3}{4}\).

Step 1: Identify Total Outcomes

The total number of possible outcomes corresponds to the total number of cards in the deck, which is 52.

Step 2: Identify Favorable Outcomes

The question asks for the probability that you are not dealt a club, therefore the favorable outcomes are any cards that are not clubs. Since a standard deck contains four suits (hearts, diamonds, clubs, spades) each having 13 cards, the number of cards that are not clubs is 52 - 13 = 39.

Step 3: Calculate Probability

The probability is obtained by dividing the number of favorable outcomes by the total number of outcomes. So the probability is \(\frac{39}{52}\), which simplifies to \(\frac{3}{4}\).

Key Concepts

Deck of Cards

A standard deck of cards is a fascinating blend of numbers, colors, and patterns. It consists of 52 cards divided evenly across four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, including three face cards (King, Queen, and Jack) and ten numbered cards from Ace (considered the 'one' card) to 10.
The deck is a common tool for teaching probability because of its uniform structure, which makes it easy to manage and comprehend.

• Hearts and Diamonds: These are the red suits.
• Clubs and Spades: These make up the black suits.

The structured nature of a deck provides clear, straightforward examples when studying probability, and it's particularly useful in exercises that require randomness and chance.

Favorable Outcomes

In probability, determining the number of favorable outcomes is crucial to understanding the likelihood of an event occurring. Favorable outcomes are those that meet the conditions specified in a particular problem. In our exercise, we want to find the probability of not drawing a club from the deck.
Here's how to identify them:

• Firstly, determine the total possible outcomes, which in this case, are the 52 cards in the deck.
• Next, calculate how many of these outcomes do not meet the specified condition (not being a club).

Since there are 13 clubs, the favorable outcomes are the cards that are not clubs, which totals 52 - 13 = 39. Understanding favorable outcomes helps pinpoint the exactly desired results from a random event.

Calculate Probability

Calculating probability is about assessing the chance of an event happening among all possibilities. This is usually expressed in fractional terms and can also be represented as decimals or percentages.
To calculate probability, use the formula:

\[P( ext{event}) = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Number of Outcomes}}\]

For our example, the probability is represented by dividing the number of non-club cards (39) by the total number of cards (52), giving \(\frac{39}{52}\).
This fraction can be simplified in some cases to make calculations easier, here resulting in \(\frac{3}{4}\).
This shows that there's a 75% chance of drawing a card that is not a club, which isn't surprising given these make up the majority of the deck.