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
$P\left(E_1\cap E_2\cap ...\cap E_6\right)$
$=P\left(E_1\right)P\left(E_2|E_1\right)....$
$=\frac{52}{52}\frac{48}{51}\frac{44}{50}\frac{40}{49}\frac{36}{48}\frac{32}{47}=0.345$