Problem 19 - Algebra for College Students

Question

One card is randomly drawn from a deck of 52 playing cards. What is the probability that it is not an ace? \(\frac{12}{13}\)

Step-by-Step Solution

Answer

The probability of drawing a card that is not an ace from a standard deck is \( \frac{12}{13} \).

Step 1: Identify the Total Number of Cards

A standard deck of playing cards contains 52 cards in total. These include cards of four suits: hearts, diamonds, clubs, and spades.

Step 2: Identify the Number of Aces

There are 4 aces in a standard deck of 52 cards. This includes one ace from each suit.

Step 3: Calculate the Probability of Drawing an Ace

The probability of drawing an ace is calculated by dividing the number of aces by the total number of cards:\[ P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \]

Step 4: Calculate the Probability of Not Drawing an Ace

The probability of an event not happening is 1 minus the probability that the event does happen. Thus, the probability of not drawing an ace is:\[ P(\text{Not Ace}) = 1 - \frac{1}{13} = \frac{12}{13} \]

Key Concepts

Probability Calculation

Probability is a measure of how likely an event is to occur. It ranges from 0 to 1, where 0 means the event is impossible, and 1 means it is certain. To calculate the probability of an event, you can use the formula:

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

Understanding this formula is key because it helps you quantify the likelihood of various outcomes.
With this calculation, you can determine the chance of drawing a specific card, winning a game, or predicting the weather. In our case, figuring out the probability of drawing an ace from a deck of cards involves counting the total favorable outcomes (aces) compared to the possible outcomes (all cards).
Mastering probability calculations will enhance your decision-making in situations involving chance.

Playing Cards Probability

When it comes to calculating probability with playing cards, it's important to know some basic facts about a standard deck. A standard deck contains 52 cards, split into four suits: hearts, diamonds, clubs, and spades, each with 13 cards.

• Each suit includes one ace, one king, one queen, and one jack, plus cards numbered 2 through 10.
• There are four aces in total in a standard deck.

To find the probability of drawing an ace, for example, divide the number of aces (favorable outcomes) by the total number of cards (possible outcomes), which is \[P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \]This calculation is foundational for understanding how likely you are to draw specific types of cards during card games.
Understanding this simple, foundational math helps you estimate the outcomes of drawing any set of cards from a deck, making you a better strategist in card games.

Event Complement Rule

The Event Complement Rule is a useful principle in probability. It revolves around finding the probability that an event does not happen. The complement of an event A is the event that A does not occur, denoted by \( A' \). Its probability is calculated using:

• \( P(A') = 1 - P(A) \)

This rule is incredibly useful for calculating probabilities indirectly. If you can easily determine the probability of an event occurring, finding the probability of it not occurring is straightforward.
In our card example, we wanted to know the probability of not drawing an ace. First, calculate the probability of drawing an ace, \[P(\text{Ace}) = \frac{1}{13} \]Then, subtract this from 1 to find \[P(\text{Not Ace}) = 1 - \frac{1}{13} = \frac{12}{13} \]This concept is particularly handy when dealing with complex probability scenarios, as it often simplifies calculations.