Question

From a pack of $52$ playing cards, three cards are drawn at random. The probability of drawing a king, queen and jack is $\dfrac{{4k}}{{5525}}$. Find the value of $k$.

Answer 1

Use the formula of combination. Find the number of favourable outcomes and total number of outcomes and then substitute the values to get the probability. We should know about the pack of cards.

Complete step-by-step solution:
The characteristic of being probable; the degree to which something is likely to happen or be the case. Probability is the branch of mathematics conserving numerical description of how likely an event is to occur. The event to which an event is likely to occur is defined as the ratio of the favourable outcomes to the total number of cases possible. There are total $52$ cards out of which $3 \times 4$ are the number of cards of King, queen, and jack.
The combination formula is written as,
${}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$
Now, the value of ${}^{52}{C_3}$ is calculated as,
${}^{52}{C_3} = \dfrac{{52!}}{{\left( {49} \right)!3!}}$
$\Rightarrow {}^{52}{C_3} = \dfrac{{52 \times 51 \times 50 \times 49!}}{{\left( {49} \right)!3!}}$
$\Rightarrow {}^{52}{C_3} = \dfrac{{52 \times 51 \times 50}}{{3 \times 2 \times 1}}$
The first $3!$ is for the order of king, queen and jack. $\dfrac{4}{{52}}$ is the probability of drawing a king out of $52$ cards ,$\dfrac{4}{{51}}$ is the probability of drawing a queen after a king and $\dfrac{4}{{50}}$ is the probability of drawing a jack once both king and queen are drawn.
As we know that the probability of drawing a king, queen and jack can be calculated as,
$P\left( E \right) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}$
$\Rightarrow P\left( E \right) = 3!{{ \times }}\dfrac{4}{{52}} \times \dfrac{4}{{51}} \times \dfrac{4}{{50}}$
$\Rightarrow P\left( E \right) = \dfrac{{16}}{{5525}}$
Now, we calculate the value of $k$ by substituting $\dfrac{{4k}}{{5525}}$ in the above equation as,
$\dfrac{{4k}}{{5525}} = \dfrac{{16}}{{5525}}$
$\therefore k = 4$

Hence, the value of $k$ is $4$.

Note: As we know that the probability is the ratio of the number of favorable outcomes to the total number of outcomes. As we know the probability is always less than or equal to one. The probability of an event in mathematics is a number between $0$ and $1$ where, $0$ indicates impossibility of the event and $1$ indicates the possibility.