The aim of this question is to find the
number of possible ways for which a $5$-
card poker hand of a total of $52 $
cards should not have
the queen of hearts.
In order to solve this problem the right way, we need to understand the concept of
Permutation and
Combination. In mathematics, the definition of
combination is the
arrangement of its given members irrespective of their order. A
permutation in mathematics is the arrangement of its members in a
definite order.
The formula of
Combination and
Permutation is as follows:
\[C\left(n,r\right)=\frac{n!}{r! \left(n-r\right)!}\]
$C\left(n,r\right)$ = number of combinations
$n$ = total number of objects
For this, we will use the formula of
combination and we have to find the
total number of possible outcomes. Then by fixing the one card from the five cards as the
queen of hearts, we have to again find the
probability with $n= 51$ and $r=4 $.
Expert Answer
According to the
rule of complement, we have:
\[ P\ (\vec{E})\ =\ 1\ -\ P(E)\]
For this we will use the formula of Combination as
the order of the members does not matter in the
set of cards containing $52 $
cards and $5$
cards were selected from the total of $52$
cards:
\[C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}\]
Here $n= 52 $ and $r = 5$ so the
formula for combination will be:
\[ C\left(52, 5\right)=\frac{ 52!}{ 5! \left( 52 – 5 \right)!} \]
Solving the above equation:
\[ C\left(52, 5\right)=\frac{ 52!}{ 5! \times 47! } \]
\[ C\left(52, 5\right) = 2,598,960 \]
Now to find the
number of hands with the
queen of hearts in it. Let us suppose we have selected
queen of hearts as
one card, then the remaining will be $4$
cards left out of the $5 $
cards to be selected from the
deck of $ 52$ cards. We have $n= 51 $ and $r = 4$ so the
formula for combination will be:
\[ C\left(51, 4\right)=\frac{ 51!}{ 4! \left( 51 – 4 \right)! } \]
Solving the above equation:
\[ C\left(51, 4\right)=\frac{ 51!}{ 4! \times 47! } \]
\[ C\left(51, 4\right) = 249,900 \]
Thus, the
probability of not finding the
queen of hearts in a
five-card poker from a total of $52$
cards will be the
possibility of the card divided by the total number of outcomes:
\[ P(E) = \dfrac {249,900 }{ 2,598,960 } \]
\[ P(E) = \dfrac {5 }{52 } \]
Now
putting the value of $P$ in the equation:
\[ P\ (\vec{E})\ =\ 1\ -\ P(E)\]
\[ P\ (\vec{E})\ =\ 1\ -\ \dfrac {5}{52}\]
\[ P\ (\vec{E})\ =\ \dfrac {47}{52}\]
\[ P\ (\vec{E})=0.9038 \]
Numerical Results
Thus, the
probability of not finding the
queen of hearts in a
five-card poker from a total of $52$
cards will be:
\[ P\ (\vec{E})\ =\ \dfrac {47}{ 52}\]
Example
A group of $4$
players is $P$, $Q$, $R$, $S$. In how many ways can a team of $2$
members be formed?
Using
the Combination formula:
$n=4$
$r=2$
\[C\left(4,2 \right)=\frac{4!}{2!\left(4-2\right)!}\]
\[C\left(4,2 \right)=6\]
