Question

The king, queen and jack of spade are removed from a deck of $52$ cards and then well shuffled. One card is selected from the remaining cards. Find the probability of getting: a club, a queen, a jack of hearts.

Answer 1

Step 1 : Use the formula of probability

$P\left(A\right)=\frac{\text{Number of outcome favorable to A}}{\text{Total number of possible outcome}}$

or

$P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}$

where,

• $P\left(A\right)$ is probability of an event $\text{'}A\text{'}$
• $n\left(A\right)$ is the favorable outcome of the event $\text{'}A\text{'}$
• $n\left(S\right)$ is total number of possible outcome in a sample space

Step 2 : Find the probability of getting a club

Let, A be the event of “getting a club card”

The king, queen and jack of spade are removed from a deck of $52$ cards

Then, Total number of cards is,

$52-3=49$

$S$ is a sample space of all the possible outcomes which is $49$ cards.

Thus, total number of possible outcome in a sample space is $n\left(S\right)=49$

Number of club card in deck of $52$ cards is $13$ cards.

Thus, the favorable outcome of the event $\text{'}A\text{'}$is $n\left(A\right)=13$

The probability of getting club card $P\left(A\right)$ is,

$P\left(A\right)=\frac{\text{number of club card}}{\text{total number of card}}=\frac{n\left(A\right)}{n\left(S\right)}$

Substitute the values on the formula,

$P\left(A\right)=\frac{13}{49}$

Step 3: Find the probability of getting a queen
Let, A be the event of “getting a queen card”
The king, queen and jack of spade are removed from a deck of $52$ cards.
Then, Total number of cards is,
$52-3=49$
$S$ is a sample space of all the possible outcomes which is $49$ cards.
Thus, total number of possible outcome in a sample space is $n\left(S\right)=49$
Number of queen card in deck of $52$ cards is $4$ cards. Already, one queen card is removed from a deck of $52$ cards. so, the remaining queen card from deck of $49$ cards is $3$ cards.
Thus, the favorable outcome of the event $\text{'}A\text{'}$ is $n\left(A\right)=3$
The probability of getting queen card $P\left(A\right)$ is,
$P\left(A\right)=\frac{\text{number of queen card}}{\text{total number of card}}=\frac{n\left(A\right)}{n\left(S\right)}$
Substitute the values on the formula,
$P\left(A\right)=\frac{3}{49}$

Step 4 : Find the probability of getting a jack of hearts

Let, A be the event of “getting a Jack of hearts”

The king, queen and jack of spade are removed from a deck of $52$cards.

Then, Total number of cards is, $52-3=49$

$S$ is a sample space of all the possible outcomes which is $49$ cards.

Thus, total number of possible outcome in a sample space is $n\left(S\right)=49$

Number of jack card in deck of $52$ cards is $4$ cards which is jack of hearts, jack of diamonds, jack of clubs, jack of spades.

Here, In a deck of cards we have only one jack of hearts card.

Thus, the favorable outcome of the event $\text{'}A\text{'}$is $1$

The probability of getting Jack of heart card$P\left(A\right)$ is,

$P\left(A\right)=\frac{\text{number of jack of hearts card}}{\text{total number of card}}=\frac{n\left(A\right)}{n\left(S\right)}$

Substitute the values on the formula,

$P\left(A\right)=\frac{1}{49}$

Therefore, the probability of getting a club $P\left(A\right)=\frac{13}{49}$,

the probability of getting a queen $P\left(A\right)=\frac{3}{49}$,

and the probability of getting a jack of hearts $P\left(A\right)=\frac{1}{49}$.