Question

Find the probability of picking a red king or black queen.

Answer 1

Hint: USe the basic information about a deck of cards as to how many different types of cards are present in a deck and solve.
Complete step by step solution:
Let A = picking a red king and B = picking a black queen. Find the probabilities of A and B add them to get the required answer. We use the fact that the intersection of A and B does not exist.

Here, the question is about a standard deck of cards.
Let us first recall the basic information about a deck of cards.

1. There are 52 cards in a standard deck.
2. There are 13 ranks of cards. They are king, queen, ace, jack, and the numbers 2 through 10.
3. There are 4 suits. They are club, spade, heart, and diamond. There are 13 cards in each suit.
4. A card is either red or black in colour. There are 26 black and 26 red cards. Diamond and heart cards are the red ones. Spades and club cards are the black ones.
5. Each rank has 4 cards i.e. one of each suit.

From the above information, we can gather that there are 4 queens - 2 red and 2 black. Similarly, there are 4 kings - 2 red and 2 black.
We need a red king or a black queen.
Now, the number of cards = 52 and the number of red cards = 26.
Also, there are 2 red kings.
Let S be the event of selecting a card out of 52 cards and A be the event of selecting a red king card out of the 2 red kings. Let B be the event where a black queen is selected.
No. of ways in which S can be done = 52
Also, no. of ways in which A can occur = 2
Therefore, the probability of selecting a red king = $P(A) = \dfrac{2}{{52}}...........(1)$
Similarly, the number of black cards = 26 and the number of black queens = 2.
Thus, the probability of selecting a black queen = $P(B) = \dfrac{2}{{52}}...........(2)$
But we need to know the probability where one selects a red king OR a black queen.
We know that$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$A \cap B$would not exist as a card can either be a red king or a black queen.
Therefore, $P(A \cap B) = 0$
Thus, the formula reduces to$P(A \cup B) = P(A) + P(B)$
This implies that the required probability is$\dfrac{2}{{52}} + \dfrac{2}{{52}} = \dfrac{4}{{52}} = \dfrac{1}{{13}}$
Hence the answer is$\dfrac{1}{{13}}$.

Note: Some students tend to skip the part where the events would have an intersection and end up with the wrong answer. For example, if the A is the event where the card is a king card and B is the event where the card is black, then $A \cap B$will not be empty as there are black king cards in a deck.