Question

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 $P\left(H^C\right)$.

b. Find $P(H^{c}andF)$. Interpret this value in context.

c. Find $P(H^{c}orF).$

Short answer

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:

$P\left(A^c\right)=P(not\hspace{0.17em}A)=1-P\left(A\right)$

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.

$P\left(H\right)=\frac{Number\hspace{0.17em}of\hspace{0.17em}favorable\hspace{0.17em}outcomes}{Number\hspace{0.17em}of\hspace{0.17em}possible\hspace{0.17em}outcomes}=\frac{13}{52}$

Usethecomplementruletohelpyou:

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

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.

$$\begin{gathered} P(H^c\hspace{0.17em}and\hspace{0.17em}F)=\frac{Number\hspace{0.17em}of\hspace{0.17em}favorable\hspace{0.17em}outcomes}{Number\hspace{0.17em}of\hspace{0.17em}possible\hspace{0.17em}outcomes} \\ =\frac{9}{52}\approx 0.1731=17.31\% \end{gathered}$$

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:

$P\left(A^c\right)=P(not\hspace{0.17em}A)=1-P\left(A\right)$

Ruleofthumbforaddition:

If there are two events,

$P(A\hspace{0.17em}or\hspace{0.17em}B)=P\left(A\right)+P\left(B\right)-P(A\hspace{0.17em}and\hspace{0.17em}B)$

We've got

Probability of not receiving a heart

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

Probability of getting a card that isn't a Heart or a Face $P(H^c\hspace{0.17em}and\hspace{0.17em}F)=\frac{9}{52}$

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 $52$.

$P\left(F\right)=\frac{Number\hspace{0.17em}of\hspace{0.17em}favorable\hspace{0.17em}outcomes}{Number\hspace{0.17em}of\hspace{0.17em}possible\hspace{0.17em}outcomes}=\frac{12}{52}$

Apply the following general addition rule:

$$\begin{gathered} P(H^c\hspace{0.17em}or\hspace{0.17em}F)=P\left(H^c\right)+P\left(F\right)-P(H^c\hspace{0.17em}and\hspace{0.17em}F)\hspace{0.17em}\hspace{0.17em} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\frac{39}{52}+\frac{12}{52}-\frac{9}{52} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\frac{42}{52}\hspace{0.17em}=\frac{21}{26}\approx 0.8077\hspace{0.17em}\hspace{0.17em}=80.77\% \end{gathered}$$

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