Question

Five cards are drawn successively with replacement from a well shuffled pack of
 $52$ cards. The probability that none is spade is ${\displaystyle \frac{243}{4^k}}$. What is the value
 of k.

Answer 1

Hint: First find the probability of the spade card and then use binomial distribution
 and solve it.

So in question it is given that Five cards are drawn successively with replacement from a well
 shuffled pack of $52$ cards. So we are given the probability of none is spade in terms of k so
 we are told to find the value of $k$.

So Let us $X$ represent the number of spade cards among the five cards drawn.
So we can see that the drawing of cards are with replacement, So the trials are Bernoulli
 trials.

A Bernoulli distribution is a Bernoulli trial. Each Bernoulli trial has a single outcome, chosen from $S$, which stands for success, or $F$, which stands for failure.

The probability of $S$ remains constant from trial-to-trial and is denoted by $p$. Write $q=1-p$for the constant probability of $F$.

The trials are independent. The probability of success is taken as p while that of failure is $q=1-p$. Consider a random experiment of items in a sale, they are either sold or not sold.
A manufactured item can be defective or non-defective. An egg is either boiled or not boiled.
A random variable $X$ will have Bernoulli distribution with probability $p$ if its probability
 distribution is

$P(X=x)=p^X{(1-p)}^X$, for $x=0,1$ and $P(X=x)=0$ for other values of $x$.
Here, $0$ is failure and $1$ is the success.

So we know, In a well shuffled pack of $52$ cards there are $13$ spade cards.
So probability of spade cards is,
$p={\displaystyle \frac{13}{52}}={\displaystyle \frac{1}{4}}$
So we know $q=1-p$
So $q=1-{\displaystyle \frac{1}{4}}={\displaystyle \frac{3}{4}}$
So $q={\displaystyle \frac{1}{4}}$
Here $X$ has a binomial distribution with $n=5$and$p={\displaystyle \frac{1}{4}}$ ,
Now $P(X=x)={}^n{c}_x{q}^{(n-x)}{p}^x$ ,where $x=0,1,2,....,n$
So for $n=5$, we get,
$P(X=x)={}^5{c}_x{\left({\displaystyle \frac{3}{4}}\right)}^{(5-x)}{\left({\displaystyle \frac{1}{4}}\right)}^x$

So we want to find probability for none is spade,
So here $x=0$
Probability for none is spade is,
$\begin{array}{rl}& P(X=0)={}^5{c}_0{\left({\displaystyle \frac{3}{4}}\right)}^{(5-0)}{\left({\displaystyle \frac{1}{4}}\right)}^0\\ & P(X=0)=1{\left({\displaystyle \frac{3}{4}}\right)}^5\\ & P(X=0)=1\left({\displaystyle \frac{243}{1024}}\right)\end{array}$
$P(X=0)={\displaystyle \frac{243}{1024}}$
So the probability that none is spade is ${\displaystyle \frac{243}{1024}}$.
So it is given that the probability of none is spade is ${\displaystyle \frac{243}{4^k}}$.
So we get,
 $\begin{array}{rl}& {\displaystyle \frac{243}{1024}}={\displaystyle \frac{243}{4^k}}\\ & {\displaystyle \frac{243}{4^5}}={\displaystyle \frac{243}{4^k}}\end{array}$
So comparing we get that, from above we can see that the value of $k=5$.

Note: You should be knowing that in $52$ cards how much different types are there.
So $52$ pack of cards contain: $13$ Diamonds, $13$ Clubs, $13$ Hearts and $13$ spades.
 Here each set of $13$ cards contain $1$ queen, king, Ace and jack. You should be knowing
 the Bernoulli trials and its formula. The formula is as follows $P(X=x)={}^n{c}_x{q}^{(n-x)}{p}^x$.