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Chapter 11: Problem 29

In Exercises 27-30, you are dealt one card from a 52-card deck Find the probability that you are dealt a heart or a picture card.

Short Answer

Expert verified
The probability of being dealt a heart or a picture card is \(\frac{5}{13}\).

Step by step solution

01

Understand a deck of cards

A standard 52-card deck has 4 suits (hearts, diamonds, clubs, spades), each containing 13 cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). Hence, there are 13 hearts and 12 picture cards (Jack, Queen, King of each suit).
02

Compute the probability of drawing a heart

The probability of an event is calculated by dividing the number of favourable outcomes by the total number of outcomes. In this case, since there are 13 hearts in a 52-card deck, the probability of drawing a heart is \(\frac{13}{52} = \frac{1}{4}\).
03

Compute the probability of drawing a picture card

There are 12 picture cards in a 52-card deck, so the probability of drawing a picture card is \(\frac{12}{52} = \frac{3}{13}\).
04

Compute the overall probability

Since these are not mutually exclusive events (a card can be a heart and a picture card at the same time), you cannot simply add the two probabilities together. To find the probability of picking either a heart or a picture card, you have to first find the probability of picking a heart that is a picture card. There are 3 hearts that are picture cards. Its probability is \(\frac{3}{52} = \frac{1}{26}\). Then you add the probabilities of getting a heart and getting a picture card then subtract the probability of getting a heart that's also a picture card. Hence, the overall probability is \(\frac{1}{4} + \frac{3}{13} - \frac{1}{26} = \frac{10}{26}\), which simplifies to \(\frac{5}{13}\).

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Key Concepts

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

Deck of Cards
A deck of cards is a complete set of playing cards used in numerous games. Most commonly, people refer to the standard 52-card deck, which includes 52 uniquely ranked and suited cards. The suits are hearts, diamonds, clubs, and spades, each containing thirteen cards. In each suit, the cards range from Ace to King, including numbered cards from 2 to 10 as well as three face cards: Jack, Queen, and King. Understanding the composition of a standard deck is key when calculating probabilities, as each suit and rank plays a role in determining the outcomes of potential card draws.
Mutually Exclusive Events
In probability, mutually exclusive events are events that cannot happen simultaneously. For instance, when tossing a coin, getting a head and a tail at the same time is impossible. However, in the context of a standard deck of cards, events like drawing a heart and drawing a picture card aren鈥檛 mutually exclusive. This means a single card can be both a heart and a picture card, such as the Jack of hearts. Recognizing whether events are mutually exclusive helps in calculating the correct probabilities. If events aren't mutually exclusive, you need to adjust for the overlap as seen in the given exercise.
Favourable Outcomes
Favourable outcomes refer to the specific outcomes that satisfy the conditions of a given event. In probability, these are the exact results that you're interested in. For example, the favourable outcomes for drawing a heart from a deck of cards are the 13 cards of the hearts suit. When calculating probability, you measure the number of favourable outcomes over the total possible outcomes. This calculation is crucial because it answers questions about likelihood and chance, such as the chance of drawing a heart or picture card defined in the exercise.
Standard 52-Card Deck
A standard 52-card deck is an everyday tool for playing card games and illustrating probability concepts. It includes evenly divided suits - hearts, diamonds, clubs, and spades, each with 13 cards. Beyond facilitating games, this deck helps explain basic probability. Each card is unique in that its suit and rank cannot be identical to another. This makes it an excellent model for individually identifying events, such as calculating the likelihood of particular cards being drawn, and teaching concepts like mutually exclusive and overlapping events.

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Most popular questions from this chapter

A construction company is planning to bid on a building contract. The bid costs the company \(\$ 1500\). The probability that the bid is accepted is \(\frac{1}{5}\). If the bid is accepted, the company will make \(\$ 40,000\) minus the cost of the bid. Find the expected value in this situation. Describe what this value means.

Write a probability problem involving the word "and" whose solution results in the probability fractions shown. \(\frac{1}{2} \cdot \frac{1}{6}\)

One card is randomly selected from a deck of cards. Find the odds in favor of drawing a black card.

An oil company is considering two sites on which to drill, described as follows: Site A: Profit if oil is found: \(\$ 80\) million Loss if no oil is found: \(\$ 10\) million Probability of finding oil: \(0.2\) Site B: Profit if oil is found: \(\$ 120\) million Loss if no oil is found: \(\$ 18\) million Probability of finding oil: \(0.1\) Which site has the larger expected profit? By how much?

The tables in Exercises 3-4 show claims and their probabilities for an insurance company. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it breaks even on its claim costs? c. How much should the company charge to make a profit of \(\$ 50\) per policy? PROBABILITIES FOR HOMEOWNERS' INSURANCE CLAIMS $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Amount of Claim (to the } \\ \text { nearest } \$ 50,000) \end{array} & \text { Probability } \\ \hline \$ 0 & 0.65 \\ \hline \$ 50,000 & 0.20 \\ \hline \$ 100,000 & 0.10 \\ \hline \$ 150,000 & 0.03 \\ \hline \$ 200,000 & 0.01 \\ \hline \$ 250,000 & 0.01 \\ \hline \end{array} $$
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