Question

Four people each draw a card from an ordinary pack. Find the chance that a card is of each suit.

Answer 1

Assume that person 1 draws a card of any suit. Now, find the probability of drawing a different suit card by person 2 considering the favourable outcomes as 39 and total outcomes as 51. For the third person consider the favourable number of outcomes as 26 and total number of outcomes as 50 and for the fourth person the total number of favourable outcomes as 13 and the total number of outcomes as 49. Find the three probabilities one by one and take their product to get the answer.

Complete step by step answer:
Here we have asked to determine the probability of choosing the four different suit cards by four persons.
Now, we know that a pack of cards contains 52 cards in total with 4 suits of cards present namely: - hearts, spades, clubs and diamonds. Each suit contains 13 cards in total. In the above question let us assume that person 1 draws a card from any particular suit. Since he is the first one to draw so he can draw any card from 52 cards so his probability of choosing a different suit card will be 1.
Therefore, person 2 must draw a card from the three remaining suits to ensure that he chose a card from a different suit. So the favourable number of outcomes for him is 39 (13 cards of 3 suits) and the total number of outcomes will be 51 since 1 card was drawn by person 1. So we get,
$\Rightarrow$ Probability for the second person = $\dfrac{39}{51}$
Similarly, the total cards now left is 50 and person 3 must draw a different suit card from the two suits remaining, so the favorable outcome is 26 and the total outcome is 50. So we get,
$\Rightarrow$ Probability for the third person = $\dfrac{26}{50}$
Now only one suit is remaining for person 4 such that he draws a different suit card that means favourable outcome is 13 and total outcome is 49. So we get,
$\Rightarrow$ Probability for the third person = $\dfrac{13}{49}$
Since all the four people must perform their functions one by one so we need to multiply the above three probabilities to get the answer.
$\Rightarrow$ Required probability = $\dfrac{39}{51}\times \dfrac{26}{50}\times \dfrac{13}{49}=\dfrac{2197}{20825}$

Note: Note that the probability of the first person to draw a different suit card doesn’t matter here because it will be 1. The reason is that he is the first one to draw the card so any card he will draw will be of a particular suit. Therefore the favourable outcomes and the total outcomes for him is 52 and so the probability becomes 1.