Question

What is the probability of getting a “FULL HOUSE” in five cards drawn in a poker game from a standard pack of $52 - Cards$? [A FULL HOUSE consists of $3$ cards of the same kind (eg, $3$ Kings) and $2$ cards of another kind (eg, $2$ Aces)]
A) $\dfrac{6}{{4165}}$
B) $\dfrac{4}{{4165}}$
C) $\dfrac{3}{{4165}}$
D) None of these

Answer 1

Hint: Anytime when you do a probability question firstly you have to write the possible outcomes and then you also have to know the total outcomes or what about you are talking about in the question. As when you know that both possible outcomes and total outcomes are the ratios of it you can consider it as your probability outcome.

Formula Used:
${ }^{n} C_{r}=\dfrac{n !}{r ! .(n-r) !}$

Complete step by step Solution:
Before starting this we just write all the possible outcomes of the given terms,
There are a total of 52 methods to select 5 cards from the deck,
By doing further solutions we get,
${ = ^{52}}{C_5} = 2598960$
As said in the question, we can take $2$ cards of the same from $4$ cards in the deck for taking it as $6$ ways.
As we know, there are $13$ different types of cards.
So, as for the total number of combinations for 2 cards is,
$6 \times 13 = 78$
As after which, there are $4$ ways to choose $3$ cards from $4$ cards of the same kind.
Since as a common that, that we know that $3$of a card suit is absolutely different from 2 of a card suit,
After which the possible combination of this is,
$4 \times 12 = 48$
So, as getting this now the total number of ways is,
$ = 48 \times 78 = 3744$
As for this, the required probability will be,
$ = \dfrac{{3744}}{{2598960}}$
After doing further solutions we get,
$ = \dfrac{6}{{4165}}$
Hence, the probability of getting a full house in five cards drawn in a poker game is
$\dfrac{6}{{4165}}$.

Therefore, the correct option is A.

Note: As we’re talking about the level of probability for the students of class 10 there is no permutation and combinations questions of probability present in class 10 but in class 11th and 12th permutation and the combination is very important for doing probability in these sessions and also very useful for doing this type of questions which was done above.