Question

If six cards are selected at random (without replacement) from a standard deck of $52$ cards, what is the probability that there will be no pairs? (two cards of same denomination)

• A. $0.28$
• B. $0.562$
• C. $0.345$
• D. $0.832$

Answer 1

Answer: (C)

Let $E_i$ be the event that the first i cards have no pair among them. Then we want to compute $P(E_6$), which is actually the same as $P(E_1\, \cap E_2\, \cap\, ...\, \cap\, E_6)$, since $E_{6} \,\subset\, E_{5}\, \subset\, ...\, \subset\, E_{1}$, implying that
$E_{1}\, \cap E_{2}\, \cap\, ...\, \cap\, E_{6} = E_{6}$.
We get
$P\left(E_{1}\, \cap E_{2}\, \cap \, ...\, \cap \, E_{6}\right)$
$ = P\left(E_{1}\right)\,P\left(E_{2} |E_{1}\right)....$
$= \frac{52}{52} \frac{48}{51} \frac{44}{50} \frac{40}{49} \frac{36}{48} \frac{32}{47} = 0.345$