Question

A deck of $52$ cards is shuffled and dealt .

Find the probabilities of the following events:

The last king appears on the ${48}^{th}$ card.

Answer 1

To find the probability that the last king appears on the ${48}^{th}$ card:

The total number of cards are $52$.

In the total cards, the number of king are $4$.

Using the concept of the probability is ratio of $\frac{\text{Event occur}}{\text{Total number of outcomes}}$, find the probability of the card is queen :

$\begin{array}{rcl}p& =& \frac{{}^4{C}_1}{{}^{52}{C}_1}\\ & \Rightarrow & p=\frac{4}{52}\\ \therefore p& =& \frac{1}{13}\end{array}$

Using the binomial distribution function $P\left(x\right)={}^{n}C_r\times p^r\times {\left(1-p\right)}^{n-r}$ and calculate the probability that the last king appears on the ${48}^{th}$ card in below :

$\begin{array}{rcl}P\left(x\right)& =& {}^{48}C_4\times {\left(\frac{1}{13}\right)}^4\times {\left(1-\frac{1}{13}\right)}^{48-4}\left[\text{Substitute all values}\right]\\ & \Rightarrow & P\left(x\right)=\frac{48!}{4!\times 44!}\times \frac{1}{{13}^4}\times {\left(\frac{12}{13}\right)}^{44}\left[\text{Simplifying}\right]\\ \therefore P\left(x\right)& =& \frac{194580\times {12}^{44}}{{13}^{48}}\end{array}$

Hence, the probability is $\begin{array}{rcl}& & \frac{\mathbf{194580}\mathbf{\times }{\mathbf{12}}^{\mathbf{44}}}{{\mathbf{13}}^{\mathbf{48}}}\end{array}$.