Question

A card is picked at random from a pack of 52 playing cards. Given that the picked card is a king, the probability of this card to be a card of spade is:

• A. 1/3
• B. 4/13
• C. 1/4
• D. 1/2

Hint:

For conditional probability problems, always clearly define and reduce your sample space first based on the "given" condition. Then, count the favorable outcomes within this reduced sample space. This simplifies complex problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks for a conditional probability: the probability that a card is a spade, given that it is a king.

Step 2: Key Formula or Approach:
Use the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Where:
- \( A \) is the event that the card is a spade.
- \( B \) is the event that the card is a king.
Alternatively, a more intuitive approach for conditional probability when the sample space is reduced:
\[ P(A|B) = \frac{\text{Number of outcomes in (A and B)}}{\text{Number of outcomes in B}} \]

Step 3: Detailed Explanation:
A standard deck of 52 playing cards consists of 4 suits (Spades, Hearts, Diamonds, Clubs), and each suit has 13 cards (A, 2, ..., 10, J, Q, K).
1. Identify the reduced sample space (Event B):
The condition is "the picked card is a king".
There are 4 kings in a deck (King of Spades, King of Hearts, King of Diamonds, King of Clubs).
So, the total number of kings = 4.
2. Identify the favorable outcome within the reduced sample space (Event A and B):
The event A is "the card is a spade".
Among the 4 kings, there is exactly one King of Spades.
So, the number of kings that are also spades = 1.
3. Calculate the conditional probability:
\[ P(\text{Spade | King}) = \frac{\text{Number of (Spade and King)}}{\text{Number of Kings}} = \frac{1}{4} \]

Step 4: Final Answer:
The probability is 1/4.