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Chapter 5: Q. 42 (page 327)

Is this your card? A standard deck of playing cards (with jokers removed) consists of 52cards in four suits鈥攃lubs, diamonds, hearts, and spades. Each suit has 13cards, 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 鈥渇ace cards.鈥 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(HC).

b. Find P(HcandF). Interpret this value in context.

c. Find P(HcorF).

Short Answer

Expert verified

a. The chance of getting non-hearts is 0.75

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

c. Non-Heart or Face cards account for approximately 80.77percent of all cards, with a probability of0.8077

Step by step solution

01

Part (a) step 1 : Given Information

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

02

Part (a) Step 2 : Simplification

Rule of complements:
P(Ac)=P(not鈥A)=1鈭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 52cards in total.
Thus, There are a total of 52possible outcomes.
Keep in mind that
Hearts make up 13of the 52cards.
Thus, There are a total of 13good outcomes. The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.
P(H)=Number鈥of鈥favorable鈥outcomesNumber鈥of鈥possible鈥outcomes=1352
Usethecomplementruletohelpyou:
P(Hc)=1鈭P(H)=1鈭1352=3952=34=0.75
Thus,
The chance of getting non-hearts is0.75
03

Part (b) step 1 : Given Information

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

04

Part (b) Step 2 : Simplification

We are aware of this.
Getting a Heart (H)
obtaining a face card (F)
Hc: acquiring a non-heart
Takealookatthetable'sbottomleftcorner.
A typical deck contains 52cards in total. Thus, There are a total of 52possible outcomes.
Keep in mind that
Face cards and non-Heart cards make for 9of the 52cards.
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(Hc鈥and鈥F)=Number鈥of鈥favorable鈥outcomesNumber鈥of鈥possible鈥outcomes=952鈮0.1731=17.31%
Thus,
Non-Heart and Face cards account for around 17.31percent of all cards, with a probability of0.1731
05

Part (c) step 1 : Given Information

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

06

Part (c) Step 2 : Simplification

Rule of complements:
P(Ac)=P(not鈥A)=1鈭P(A)
Ruleofthumbforaddition:
If there are two events,
P(A鈥or鈥B)=P(A)+P(B)鈭P(A鈥and鈥B)
We've got
Probability of not receiving a heart
P(Hc)=3952
Probability of getting a card that isn't a Heart or a Face P(Hc鈥and鈥F)=952
Now, Keep in mind that
Face cards make up 12of the 52cards.
Thus,
The number of positive outcomes in this scenario is 12while the number of alternative outcomes is 52.
P(F)=Number鈥of鈥favorable鈥outcomesNumber鈥of鈥possible鈥outcomes=1252
Apply the following general addition rule:

P(Hc鈥or鈥F)=P(Hc)+P(F)鈭P(Hc鈥and鈥F)鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥=3952+1252鈭952鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥=4252鈥=2126鈮0.8077鈥鈥=80.77%

Non-Heart or Face cards account for approximately 80.77percent of all cards, with a probability of0.8077

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