Question

Five card are drawn successively from a pack of $52$ cards with replacement. Find the probability that there is at least one Ace

• A. $\frac{1}{(13{)}^5}$
• B. $\frac{\left(\right(12{)}^5}{(13{)}^5}$
• C. $\frac{(13{)}^5-(12{)}^5}{(13{)}^5}$
• D. $\frac{1}{(12{)}^5}$

Answer 1

The correct option is C $\frac{(13{)}^5-(12{)}^5}{(13{)}^5}$

Given, five cards are drawn successively from a pack of $52$ cards with replacement.
The probability there is atleast one ace $=1-$ probability that none of the cards drawn is ace.
$=1-\left(\frac{48}{52}\times \frac{48}{52}\times \frac{48}{52}\times \frac{48}{52}\times \frac{48}{52}\right)$
(since in a pack of $52$ cards, there will be $48$ non-ace cards)$=\frac{({13}^5-{12}^5)}{{13}^5}$