Problem 1 - Discrete Mathematics with Applications

Question

A card is drawn at random from a standard deck of cards. Find the probability of obtaining: A king.

Step-by-Step Solution

Answer

The probability of drawing a king from a standard deck of cards is \(\frac{1}{13}\).

Step 1: Determine the number of successful outcomes

There are 4 kings in a standard deck of cards (one king per suit: hearts, diamonds, clubs, and spades). So, there are 4 successful outcomes.

Step 2: Determine the total number of possible outcomes

There are 52 cards in a standard deck of cards. So there are 52 possible outcomes when drawing a card at random.

Step 3: Calculate the probability

To find the probability of drawing a king, we will divide the number of successful outcomes (4 kings) by the total number of possible outcomes (52 cards): \[P(King) = \frac{Number\:of\:successful\:outcomes}{Total\:number\:of\:possible\:outcomes}\] \[P(King) = \frac{4}{52}\]

Step 4: Simplify the fraction

We can simplify the fraction \(\frac{4}{52}\) by finding the greatest common divisor of 4 and 52, which is 4. Dividing both numerator and denominator by 4, we get: \[P(King) = \frac{4 \div 4}{52 \div 4} = \frac{1}{13}\] So, the probability of drawing a king from a standard deck of cards is \(\frac{1}{13}\).

Key Concepts

Probability Calculation

Let's dive into the world of probability calculation, a fundamental concept in mathematics that deals with the likelihood of an event occurring. Probability can be expressed as a number between 0 and 1, where 0 indicates impossibility and 1 denotes certainty.

In the scenario of drawing a card from a standard deck, we're looking to find out how likely it is to draw a king. To do that, we start by setting up a ratio. The success, which in this case is drawing a king, is compared to all the possible outcomes, which includes drawing any card from the deck.

Setting Up the Probability Ratio

When deciding the probability of drawing a king, we set it up as:
\[P(King) = \frac{Number\:of\:successful\:outcomes}{Total\:number\:of\:possible\:outcomes}\]
This probability ratio forms the cornerstone of our understanding. By assigning a quantitative value to our chances, we transform the abstract concept of 'likelihood' into a concrete number that can be analyzed and understood.

Successful Outcomes in Probability

In the context of probability, a 'successful outcome' refers to an outcome that fulfills the conditions of the event we are interested in.

For instance, in our card-drawing exercise, a successful outcome is drawing one of the four kings in the deck. It's crucial to identify all the different ways success can be achieved in any given scenario as this defines the numerator in our probability calculation.

Counting Successes

In a standard deck of 52 cards, there are exactly 4 kings. Each of these represents a unique successful outcome. By having a clear definition of what counts as a success, we can confidently say there are 4 successful outcomes out of the total 52 cards. This is the basis for our probability calculation and is essential for understanding the odds of random events.

Simplifying Fractions

Now, let's tackle the process of simplifying fractions, which is a helpful technique to present probability in its simplest form. Simplification makes understanding and comparing probabilities much easier.

In the given problem, after calculating the initial probability of drawing a king, we arrived at a fraction: \(\frac{4}{52}\). To make this fraction simpler and thus easier to understand, we look for the greatest common divisor of the numerator and denominator.

Making Fractions Understandable

The greatest common divisor of 4 and 52 is 4, so we can divide both the top and bottom numbers by 4 to simplify the fraction:
\[\frac{4 \div 4}{52 \div 4} = \frac{1}{13}\]
With the fraction simplified, it becomes immediately more comprehensible. In the context of our card example, a simplified fraction translates into a clearer understanding that out of every 13 cards, statistically, one should be a king.