Problem 38 - College Algebra

Question

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

Step-by-Step Solution

Answer

The probability of not being dealt a picture card is \(\frac{40}{52}\), simplified to \(\frac{10}{13}\) when reduced to the lowest terms.

Step 1: Determine Total Number of Outcomes

There are a total of 52 possible outcomes when drawing a card from a deck, as a deck of cards consists of 52 cards.

Step 2: Calculate Favorable Outcomes for Drawing a Picture Card

There are 3 picture cards per suit (jack, queen, and king) and 4 suits in total. So there are \(3 \times 4 = 12\) picture cards in a deck.

Step 3: Calculate Probability of Drawing a Picture Card

The probability \(P(A)\) of an event \(A\) is given as the number of favorable outcomes divided by the total number of outcomes. So, the probability of drawing a picture card is \(\frac{12}{52}\).

Step 4: Find Complement of the Probability

The complement of an event A, denoted by \(A^c\), is the event not occurring. The probability of the complement of an event is \(1 - P(A)\). So, \(1- \frac{12}{52} = \frac{40}{52}\).

Key Concepts

Deck of Cards

A standard deck of cards is a fascinating tool often used in probability exercises due to its structured yet variable nature. It consists of 52 individual cards. These cards are divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards starting from Ace

• Number cards: 2 to 10
• Picture cards: Jack, Queen, and King
• An Ace

This setup creates a total of 52 possible outcomes when drawing one random card from the deck, which forms the basis for calculating probabilities in such scenarios. Understanding the composition of the deck is crucial for determining how likely different outcomes are, such as drawing specific cards or groups of cards. This uniformity in each category of cards allows us to systematically explore probabilities by easily categorizing cards into favorable and unfavorable outcomes.

Complementary Probability

In probability, there's a neat trick known as complementary probability that helps calculate how likely we are to not get a certain result. When focusing on the probability of an event not occurring, we refer to its complement, denoted as \(A^c\). For example, if you're trying to find the probability that you are not dealt a picture card from a deck, complementary probability is quite helpful.

• First, identify the probability of the event itself occurring (in this case, getting a picture card).
• Then, subtract this probability from 1 to get the probability of the complement (not getting a picture card).

Mathematically, if the probability of the event \(A\) happening is \(P(A)\), then the probability of the complement \(A^c\) is \(1 - P(A)\). So, even if directly calculating a probability seems tricky, complementary probability offers a simpler path by leveraging known probabilities to find unknown ones, like figuring out what happens in all other possible outcomes.

Favorable Outcomes

The concept of favorable outcomes is central to understanding probability. It means determining how many ways an event can occur in a way that is desired or satisfies the condition being studied. In the context of card decks, if you want to find the probability of not drawing a picture card, you first need to figure out how many pictures there are.

• A standard deck has 12 picture cards: 3 Jack, Queen, King per each of the 4 suits.
• Hence, portraying a favorable outcome as not getting a picture card means looking at the remaining non-picture cards.

Once the scenario for your target outcome is set up, probability calculation involves dividing these favorable outcomes by the total possible outcomes. In this example, there are 40 cards that are not picture cards (52 total cards minus the 12 picture cards), establishing a scenario where 40 favorable outcomes exist. This streamlines the process of computational probability by separating desirable outcomes from the total set of possibilities, effectively framing the probability equation needed.