Chapter 5: Q. 42 (page 327)

Is this your card? A standard deck of playing cards (with jokers removed) consists of $52$ cards in four suits—clubs, diamonds, hearts, and spades. Each suit has $13$ cards, with denominations ace, $2, 3, 4, 5, 6, 7, 8, 9, 10,$ jack, queen, and king. The jacks, queens, and kings are referred to as “face cards.” Imagine that we shuffle the deck thoroughly and deal one card. Define events $F$ : getting a face card and $H$ : getting a heart. The two-way table summarizes the sample space for this chance process
a. Find $P (H^{C})$ .
b. Find $P (H^{c} a n d F)$ . Interpret this value in context.
c. Find $P (H^{c} o r F) .$

Short Answer

Expert verified

a. The chance of getting non-hearts is $0.75$

b. Non-Heart and Face cards account for around $17.31$ percent of all cards, with a probability of $0.1731$

c. Non-Heart or Face cards account for approximately $80.77$ percent of all cards, with a probability of $0.8077$

Step by step solution

Part (a) step 1 : Given Information

We have to determine the probability for getting non − hearts.

Part (a) Step 2 : Simplification

Rule of complements:

P (A^{c}) = P (n o t A) = 1 - P (A)

We are aware of this.
Getting a Heart (

H

) and obtaining a face card (

F

)

Take a look at the table's bottom left corner.

A typical deck contains

52

cards in total.

Thus, There are a total of

52

possible outcomes.

Keep in mind that

Hearts make up

13

of the

52

cards.

Thus, There are a total of

13

good outcomes. The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

P (H) = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{N u m b e r o f p o s s i b l e o u t c o m e s} = \frac{13}{52}

Usethecomplementruletohelpyou:

P (H^{c}) = 1 - P (H) = 1 - \frac{13}{52} = \frac{39}{52} = \frac{3}{4} = 0.75

Thus,

The chance of getting non-hearts is

0.75

Part (b) step 1 : Given Information

We have to determine the probability for getting non − hearts and face card.

Part (b) Step 2 : Simplification

We are aware of this.

Getting a Heart (

H

)

obtaining a face card (

F

)

H^{c}

: acquiring a non-heart

Takealookatthetable'sbottomleftcorner.

A typical deck contains

52

cards in total. Thus, There are a total of

52

possible outcomes.

Keep in mind that

Face cards and non-Heart cards make for

9

of the

52

cards.

Thus,

There are nine positive outcomes.

The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

P (H^{c} a n d F) = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{N u m b e r o f p o s s i b l e o u t c o m e s} = \frac{9}{52} \approx 0.1731 = 17.31 %

Thus,

Non-Heart and Face cards account for around

17.31

percent of all cards, with a probability of

0.1731

Part (c) step 1 : Given Information

We have to determine the probability for getting non − heart or face card.

Part (c) Step 2 : Simplification

Rule of complements:

P (A^{c}) = P (n o t A) = 1 - P (A)

Ruleofthumbforaddition:

If there are two events,

P (A o r B) = P (A) + P (B) - P (A a n d B)

We've got

Probability of not receiving a heart

P (H^{c}) = \frac{39}{52}

Probability of getting a card that isn't a Heart or a Face

P (H^{c} a n d F) = \frac{9}{52}

Now, Keep in mind that

Face cards make up

12

of the

52

cards.

Thus,

The number of positive outcomes in this scenario is

12

while the number of alternative outcomes is

52

P (F) = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{N u m b e r o f p o s s i b l e o u t c o m e s} = \frac{12}{52}

Apply the following general addition rule:

$P (H^{c} o r F) = P (H^{c}) + P (F) - P (H^{c} a n d F) = \frac{39}{52} + \frac{12}{52} - \frac{9}{52} = \frac{42}{52} = \frac{21}{26} \approx 0.8077 = 80.77 %$

Non-Heart or Face cards account for approximately

80.77

percent of all cards, with a probability of

0.8077

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Part (a) step 1 : Given Information

Part (a) Step 2 : Simplification

Part (b) step 1 : Given Information

Part (b) Step 2 : Simplification

Part (c) step 1 : Given Information

Part (c) Step 2 : Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Statistics

Decision Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.