The probability of drawing a king and a queen from a deck of 52 cards is [tex]\boxed{\frac{8}{{663}}}.[/tex]
Further explanation:
The formula of combination can be expressed as,
[tex]\boxed{^n{C_r} = \frac{{n!}}{{r!\left( {n - r} \right)!}}}[/tex]
The probability can be defined as the ratio of favorable number outcomes to the total number of outcomes.
[tex]\boxed{{\text{Probability}} = \frac{{{\text{Favorable number of outcomes}}}}{{{\text{Total number of outcomes}}}}}[/tex]
Given:
A deck contains 52 playing cards.
Calculation:
Two cards are drawn from a deck of 52 cards.
The number of total possible outcomes can be obtained as,
[tex]{\text{Total Possible}}{\kern 1pt} {\text{ outcomes}} = {{\kern 1pt} ^{52}}{C_2}[/tex]
Substitute 52 for n and 2 for r in equation [tex]{^n{C_r} = \frac{{n!}}{{r!\left( {n - r} \right)!}}}[/tex]) to obtain the possible outcomes.
[tex]\begin{aligned}{\text{Possible }}{\kern 1pt} {\text{outcomes}}&={{\kern 1pt} ^{52}}{C_2} \\&= \frac{{52!}}{{2!\left( {52 - 2} \right)!}} \\ &= \frac{{52!}}{{50!\left( {2!} \right)}} \\ &= \frac{{52 \times 51 \times 50!}}{{50!{\kern 1pt} \times 2 \times 1}} \\ & = 1326 \\ \end{aligned}[/tex]
There are 4 kings and 4 queens in a deck of 52 cards.
Favorable number of outcomes can be obtained as,
[tex]\begin{aligned} {\text{Favorable number of outcomes}} &= {}^4{C_1} \times {}^4{C_1} \\ &= 4 \times 4 \\ & = 16 \\ \end{aligned}[/tex]
The probability of drawing a king and a queen from a deck of 52 cards can be obtained as,
[tex]\begin{aligned} {\text{Probability}} &= \frac{{16}}{{1326}} \\ &= \frac{8}{{663}} \\ \end{aligned}[/tex]
Hence, the probability of drawing a king and a queen from a deck of 52 cards is [tex]\boxed{\frac{8}{{663}}}.[/tex]
Learn more:
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Probability
Keywords: picks, king, queen, possible, outcomes, select, combination, randomly picks, a deck contains 52 cards, aces, jacks.
