Question

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

Find the probabilities of the following events:

The tenth card is a queen.

Answer 1

To find the probability that the tenth card is a queen:

The total number of cards are $52$.

In the total cards, there are the number of queens 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 tenth card is a queen in below :

$\begin{array}{rcl}P\left(x\right)& =& {}^{10}C_1\times {\left(\frac{1}{13}\right)}^1\times {\left(1-\frac{1}{13}\right)}^{10-1}\left[\text{Substitute all values}\right]\\ & \Rightarrow & P\left(x\right)=10\times \frac{1}{13}\times {\left(\frac{12}{13}\right)}^9\left[\text{Simplifying}\right]\\ \therefore P\left(x\right)& =& \frac{10\times {12}^9}{{13}^{10}}\end{array}$

Hence, the probability is $\begin{array}{l}\frac{\mathbf{10}\mathbf{\times }{\mathbf{12}}^{\mathbf{9}}}{{\mathbf{13}}^{\mathbf{10}}}\end{array}$.