Multiple choice

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

  1. $\displaystyle \frac {1}{(13)^5}$
  2. $\displaystyle \frac {((12)^5}{(13)^5}$
  3. $\displaystyle \frac {(13)^5-(12)^5}{(13)^5}$
  4. $\displaystyle \frac {1}{(12)^5}$
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C Correct answer
Explanation

With replacement, the probability of drawing an Ace on one draw is 1/13, so the probability of no Ace is 12/13. The probability of at least one Ace in five draws is therefore 1 - (12/13)^5, equal to ((13)^5 - (12)^5)/(13)^5.

AI explanation

The probability of drawing an Ace from a standard 52-card deck is 4/52, which simplifies to 1/13. The probability of not drawing an Ace in a single trial is therefore 12/13. Using the complement rule, the probability of getting at least one Ace in five independent draws is 1 minus the probability of getting no Aces in five draws. This calculates to 1 minus (12/13)^5, which can be written as (13^5 - 12^5) divided by 13^5.

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