Question

In bridge games of playing cards, 4 players are distributed one card each by turn so that each player gets 13 cards. Find out the probability of a specified player getting a black ace and a king.
A) $p = \dfrac{{82251}}{{978775}}$
B) $p = \dfrac{{164502}}{{978775}}$
C) $p = \dfrac{{329004}}{{978775}}$
D) $p = \dfrac{{82251}}{{1957550}}$

Answer 1

Here, we will use the formula of combination to find out the number of ways of selecting 13 cards out of the total deck of 52 cards. Then we will find the number of ways of selecting a black ace, a king and all the remaining cards using the formula of combination. By dividing the favorable outcomes by the total number of outcomes, we will get the required probability.

Formula Used:
We will use the following formula:
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ , where $n$ is the total number of elements and $r$ is the number of elements to be selected
Probability $=$ Number of favorable outcomes $\div$ Total number of outcomes

Complete Step by Step Solution:
According to the question, it is given that in bridge games of playing cards, 4 players are distributed one card each by turn so that each player gets 13 cards.
Therefore, total number of cards which each player can get$= 13$
Now, we know that the total number of cards is 52. Therefore
The number of ways of selecting 13 cards out of the total cards $= {}^{52}{C_{13}}$
Using the formula ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ , we get
$\Rightarrow$ The number of ways of selecting 13 cards out of the total cards $= \dfrac{{52!}}{{13!\left( {52 - 13} \right)!}}$
Subtracting the terms in the denominator, we get
$\Rightarrow$ The number of ways of selecting 13 cards out of the total cards $= \dfrac{{52!}}{{13!39!}}$ ………………..$\left( 1 \right)$
Now, we know that we have 2 back aces in a deck of 52 cards.
So, the probability of selecting 1 black ace out of two is ${}^2{C_1}$.
Similarly, there are 4 kings, so
The probability of selecting 1 king out of 4 is ${}^4{C_1}$.
Now we will find the remaining i.e. the cards left after removing these 4 kings and 2 black ace out of a deck of 52 cards.
Remaining cards $= 52 - 4 - 2 = 46$ cards
Also, after choosing 1 king and 1 black ace we are required to select only $12 - 2 = 11$cards out of these 46 cards
Thus, the combination required will be ${}^{46}{C_{11}}$.
Hence, total number of ways of selecting 13 cards out of the deck of 52 cards $= {}^{52}{C_{13}}$
And, the favorable outcomes of selecting those 13 cards $= {}^2{C_1} \times {}^4{C_1} \times {}^{46}{C_{11}}$
Hence, the required probability is the fraction of number of favorable outcomes by total number of outcomes $= \dfrac{{{}^2{C_1} \times {}^4{C_1} \times {}^{46}{C_{11}}}}{{{}^{52}{C_{13}}}}$
Solving the combination and substituting the value from equation $\left( 1 \right)$, we get,
Probability, $p = \dfrac{{2 \times 4 \times \dfrac{{46!}}{{11!35!}}}}{{\dfrac{{52!}}{{13!39!}}}} = \dfrac{{\dfrac{{8 \times 46!}}{{11!35!}}}}{{\dfrac{{52!}}{{13!39!}}}}$
$\Rightarrow p = \dfrac{{\dfrac{{8 \times 46!}}{{11!35!}}}}{{\dfrac{{52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46!}}{{13 \times 12 \times 11! \times 39 \times 38 \times 37 \times 36 \times 35!}}}}$
Simplifying the expression, we get
$\Rightarrow p = \dfrac{{8 \times 13 \times 12 \times 39 \times 38 \times 37 \times 36}}{{52 \times 51 \times 50 \times 49 \times 48 \times 47}}$
Cancelling the similar terms, we get
$\Rightarrow p = \dfrac{{13 \times 3 \times 19 \times 37 \times 6}}{{17 \times 25 \times 49 \times 47}} = \dfrac{{164502}}{{978775}}$
Therefore, the probability of a specified player getting a black ace and a king is: $p = \dfrac{{164502}}{{978775}}$

Hence, option B is the correct answer.

Note:
A bridge game means a game of four people where two people play against the other two as partners. A standard 52 deck play card is used and it is dealt out one at a time in a clockwise manner round the table so that each player gets 13 cards in total. To answer this question, it is really important to know that there are 52 cards in total out of which 26 are red in colour and 26 are black. Now, the red cards are divided into hearts and diamonds of 13 each. Whereas, the black cards are divided into spades and clubs of 13 each. There are total 12 face cards i.e. King, Queen and Jack and each suit consists of 3 face cards each.