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It has been given that two decks of cards are 鈥渕atched,鈥� that is, the order of the cards in the decks is compared by turning the cards over one by one from the two decks simultaneously; a 鈥渕atch鈥� means that the two cards are identical

Probability means the chances of any event to occur is called it probability.

The Poisson distribution is given by the expression mentioned below.

$P\left(x\right)=\frac{{渭}^x鈰�e^{鈭�渭}}{x!}$

Assume that there is no cards match so the average.

$\begin{array}{c}渭=np\\ =n\left(\frac{1}{n}\right)\\ =1\end{array}$

All couples are equally likely, the probability of getting no match is given by expression mentioned below.

$\begin{array}{c}p\left(0\right)=\frac{1^0鈰�e^{鈭�1}}{0!}\\ =e^{鈭�1}\end{array}$

Now the probability of getting at least one couples is given by expression mentioned below.

$\begin{array}{c}p(x>0)=1鈭�p\left(0\right)\\ =1鈭�e^{鈭�1}\\ =1鈭�\frac{1}{e}\end{array}$

Hence, it has been proved.