Problem 3 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The probability of not being dealt a heart is \(39/52\) or approximately 0.75 when rounded to two decimal places.

Step 1: Identify the Total Number of Outcomes

In a standard 52 cards deck, the total number of possible outcomes when being dealt one card is 52, since there are 52 cards in total.

Step 2: Find Number of Favorable Outcomes

Each suit (hearts, diamonds, clubs and spades) has exactly 13 cards in a deck. So, there are 13 hearts in the deck. Since we are interested in not getting a heart, the favorable outcomes will be total outcomes minus the outcomes of getting a heart, which is \(52 - 13 = 39\). These are the outcomes where a heart is not dealt.

Step 3: Calculate the Probability

Now, we can calculate the probability. The probability (P) of an event happening is the number of ways that event can happen (favorable outcomes) divided by the total number of outcomes. So the probability of not dealing a heart will be \(P = 39/52\).

Key Concepts

Card deck

A standard card deck consists of 52 cards. These cards are divided equally into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 unique cards, ranging from an ace to a king. Understanding how these cards are organized is crucial for tackling probability problems related to card games. Every time you draw a card from a deck, without any prior knowledge of what card it may be, you have an equal chance of drawing any one of the 52 cards. This fixed number of cards is essential when calculating probability.

Mathematical outcomes

When dealing with probability, every event has possible outcomes. For a card deck, each card draw represents a single outcome. In our exercise, we're interested in outcomes that do not result in drawing a heart. Here, the mathematical outcome comprises two parts: favorable outcomes and total possible outcomes. The total outcomes, in this case, are the total 52 cards in the deck. Favorable outcomes represent situations aligning with our condition, which are the 52 cards minus the 13 hearts. Thus, the focus is on the remaining 39 cards which are neither hearts.

Probability calculation

Probability involves calculating the likelihood of an event occurring. The basic formula for probability is:

• Probability (P) = Number of Favorable Outcomes / Total Number of Outcomes

In our scenario, we're asked to find the probability of not drawing a heart from a deck of cards. Knowing that there are 52 cards in total and 13 of them are hearts means the favorable outcomes number 39. Therefore, the calculation you perform is:\[ P = \frac{39}{52} \]This equation provides the probability of not being dealt a heart. In simplified terms, you have a 39 out of 52 chance to draw any card other than a heart.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations. It is essential in probability for calculating outcomes. Although our example here is straightforward and does not require advanced combinatorics, understanding its basics helps in more complex probability scenarios.

• Combinations dictate how many ways you can choose items where the order doesn't matter.
• Permutations consider how many ways you can arrange items where the order does matter.

In problems involving card decks, combinatorics often comes into play when you're dealing more than one card, or when considering ordered sequences of cards. Although for our current problem of drawing a single card a simple subtraction suffices, combinatorics provides the framework for expanded and more complex probability calculations.