/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 You are dealt one card from a st... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 19

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a picture card.}$$

Short Answer

Expert verified
The probability of being dealt a picture card from a standard deck is approximately \(0.231\) or \(\frac{12}{52}\) in fraction form.

Step by step solution

01

Count total outcomes

Count the total number of cards in the deck. In a standard deck, there are 52 cards.
02

Count successful outcomes

Count the total number of picture cards (jacks, queens, kings) in the deck. There are 4 each of jacks, queens, and kings, making 12 picture cards in total.
03

Calculate probability

Divide the number of successful outcomes (12) by the total number of outcomes (52). The probability is therefore \(\frac{12}{52}\) or approximately 0.231 when rounded to three decimal places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides a quantifiable measure to the concept of 'chance,' which is fundamental in understanding how likely it is for a particular event to happen. When calculating probability, the formula often used is \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \), where \( P(E) \) represents the probability of event \( E \) happening.

It's important for students to recognize that the probabilities are always between 0 and 1, inclusive. A probability of 0 means the event will never occur, while a probability of 1 indicates certainty of the event. Values between 0 and 1 indicate the event's likelihood in relation to all possible outcomes. When dealing with a finite number of equally likely outcomes, such as drawing cards from a deck, probability theory becomes a powerful tool to anticipate the frequency of certain events.
Standard 52-Card Deck
A standard 52-card deck is a common collection of playing cards used in various games. Each deck contains four suits: hearts, diamonds, clubs, and spades. Each suit is made up of 13 cards, which include numbered cards from 2 to 10, an ace, and three picture cards鈥攖he Jack, Queen, and King.

Understanding the composition of the deck is crucial in many card-based probability problems. With this knowledge, one can calculate the probability of events, such as being dealt a particular type of card. Since each suit has the same number of cards and the same ranking, the 52-card deck is a perfect example of a uniformly distributed set of items, which simplifies probability calculations.
Picture Cards in Deck
Picture cards, also known as face cards, are the Jack, Queen, and King in each suit of a standard 52-card deck. Each of these ranks has one card in each of the four suits, totaling twelve picture cards in the entire deck. This is important for probability because the more picture cards or any specific group of cards there are, the greater the chances of drawing one from the deck.

For instance, the odds of pulling any picture card can be calculated by dividing the total number of picture cards (12) by the total number of cards (52). This yields the probability of approximately 0.231. It is helpful for students to visualize the deck and remember that there are three picture cards per suit, as this can aid in their understanding of probability concepts and calculations.

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Most popular questions from this chapter

Explaining the Concepts The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user. Hint: Find the following probability fraction: the number of employees who test positive and are cocaine users the number of employees who test positive This fraction is given by $$ 90 \% \text { of } 1 \% \text { of } 10,000 $$ the number who test positive who actually use cocaine plus the number who test positive who do not use cocaine What does this probability indicate in terms of the percentage of employees who test positive who are not actually users? Discuss these numbers in terms of the issue of mandatory drug testing. Write a paper either in favor of or against mandatory drug testing, incorporating the actual percentage accuracy for such tests.

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n} .\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{M M M, M M F, M F M, M F F, F M M FMF, FFM, FFF\)} - Find the probability of selecting a family with $$\text{at least one male child.}$$

A new factory in a small town has an annual payroll of S6 million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?

The table shows the population of Texas for 2000 and 2010 with estimates given by the U.S. Census Bureau for 2001 through 2009 \(\begin{array}{llllll}{\text { Year }} & {2000} & {2001} & {2002} & {2003} & {2004} \\ \hline \text { Population } & {20.85} & {21.27} & {21.70} & {22.13} & {22.57} & {23.02} \\\ \hline\end{array}\) \(\begin{array}{llllll}{\text { Year }} & {2006} & {2007} & {2008} & {2009} & {2010} \\ \hline \text { Population } & {23.48} & {23.95} & {24.43} & {24.92} & {25.15} \\ {\text { in millions }} & {23.48} & {23.95} & {24.43} & {24.92} & {25.15}\end{array}\) a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after 1999 c. Use your model from part (b) to project Texas's population, in millions, for the year \(2020 .\) Round to two decimal places.
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