Question

Two cards are drawn from a pack of $52$ cards. Find the probability distribution of
(i) $X=$ number of picture cards
(ii) $Y=$ number of black cards
(iii) $Z=$ number of diamond cards.

Answer 1

In this problem we have to calculate the probability distribution of the given cards. In the question they have mentioned that two cards are drawn from a pack of $52$, so we will take the number of ways to draw two cards from $52$ cards. For each case we will write the number of cards we have and the number ways to draw two cards. Now we will take the ratio of both the values, to get the required probability.

Complete Step by Step Solution:
Given that, two cards are drawn from the pack of $52$ cards.
We can take two cards from $52$ cards in ${}^{52}{{C}_{2}}$ ways.
Considering the first case.
In this case they have mentioned picture cards.
In a pack of $52$ we have $12$ picture cards.
We can take two cards from $12$ picture cards in ${}^{12}{{C}_{2}}$.
Now the probability of the first case is
$P\left( X \right)=\dfrac{{}^{12}{{C}_{2}}}{{}^{52}{{C}_{2}}}$
Considering the second case.
In this case they have mentioned about black cards.
In a pack of $52$ we have $26$ black cards.
We can take two cards from $26$ picture cards in ${}^{26}{{C}_{2}}$.
Now the probability of the first case is
$P\left( Y \right)=\dfrac{{}^{26}{{C}_{2}}}{{}^{52}{{C}_{2}}}$.
Considering the third case.
In this case they have mentioned diamond cards.
In a pack of $52$ we have $13$ black cards.
We can take two cards from $13$ picture cards in ${}^{13}{{C}_{2}}$.
Now the probability of the first case is
$P\left( Y \right)=\dfrac{{}^{13}{{C}_{2}}}{{}^{52}{{C}_{2}}}$.

Note:
For the problems related to playing cards, we need to have some clarity about the number of different cards in the pack. The typical pack is given by