Question

From a pack of $52$ cards $4$ are drawn one by one without replacement, find the probability that all are aces.

Answer 1

The number of aces in a pack of 52 card is 4, so the probability that the card drawn is an ace is,

$P=\frac{4}{52}$

Now, since one card is been chosen, then the total number of aces left is 3 and the total number of cards left is $52-1=51$.

The probability that the card drawn is an ace is,

$P=\frac{3}{51}$

As two cards are been chosen, then the total number of aces left is 2 and the total number of cards left is $51-1=50$.

The probability that the card drawn is an ace is,

$P=\frac{2}{50}$

Again, since three cards are been chosen, then the total number of aces left is 1 and the total number of cards left is $50-1=49$.

The probability that the card drawn is an ace is,

$P=\frac{1}{49}$

By multiplication law of probability,

$\frac{4}{52}\times \frac{3}{51}\times \frac{2}{50}\times \frac{1}{49}=\frac{24}{6497400}$

$=\frac{1}{270725}$

Therefore the probability that all the cards are aces is $\frac{1}{270725}$.