Question

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

Answer 1

Hint: we can check the probability of aces when 4 cards are drawn using:
\[probability{\text{ }} = {\text{ }}\dfrac{{favorable{\text{ }}outcomes}}{{total{\text{ }}outcomes}}\]
Where the favorable outcomes will be the number of ace cards and the total outcome will be the total number of cards.
When the cards are chosen without replacement, with each card chosen, both the cards as favorable outcomes and total outcomes are decreased by 1.

Complete step-by-step answer:
Total cards in a deck = 52
Number of ace cards in deck = 4
Using \[\left[ {probability{\text{ }} = {\text{ }}\dfrac{{favorable{\text{ }}outcomes}}{{total{\text{ }}outcomes}}} \right]\]:
i) Probability that the first card drawn is Ace:
Favorable outcomes ( ace cards ) = 4
Total outcomes ( total cards ) = 52
${P_1} = \dfrac{4}{{52}}$
Now, the cards are chosen without replacement, this means every time an ace card is chosen the total cards and number of ace cards are reduced by 1.
ii) Probability that the second card drawn is Ace:
Favorable outcomes ( ace cards ) 🡪 4 – 1 = 3
Total outcomes ( total cards ) 🡪 52 – 1 = 51
${P_2} = \dfrac{3}{{51}}$
iii) Probability that the third card drawn is Ace:
Favorable outcomes ( ace cards ) 🡪 3– 1 = 2
Total outcomes ( total cards ) 🡪 51 – 1 = 50
${P_3} = \dfrac{2}{{50}}$
ii) Probability that the fourth card drawn is Ace:
Favorable outcomes ( ace cards ) 🡪 2 – 1 = 1
Total outcomes ( total cards ) 🡪 50 – 1 = 49
${P_4} = \dfrac{1}{{49}}$
The required probability is given as:
$P = {P_1} \times {P_2} \times {P_3} \times {P_4}$
Substituting their calculated values from i), ii), iii) and iv), we get:
   $P = \dfrac{4}{{52}} \times \dfrac{3}{{51}} \times \dfrac{2}{{50}} \times \dfrac{1}{{49}}$
$
  P = \dfrac{{24}}{{6497400}} \\
  P = \dfrac{1}{{270725}} \\
 $
Therefore, from a pack of 52 cards if 4 are drawn one by one without replacement, the probability that all are aces is $\dfrac{1}{{270725}}$ .

Note: In case of no replacement, the number of cards reduces by 1 because the already chosen card does not get mixed in the deck again and hence won’t be counted for the next card while choosing.
Ace refers to the card ‘A’ and each suite has one ace. As there are 4 suites, the number of ace present in the deck is also 4.