This problem involves calculating a conditional probability, specifically the probability of drawing a third king from a standard deck of 52 cards, given that the first two cards drawn were kings. This scenario highlights the concept of dependent events, where the outcome of prior draws affects the possibilities for subsequent draws because cards are not replaced.

Cards Deck Composition

To solve this probability problem, it's crucial to understand the composition of a standard pack of 52 cards:

• Total number of cards: 52
• Number of suits: 4 (Hearts, Diamonds, Clubs, Spades)
• Number of cards in each suit: 13 (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King)
• Number of King cards in a full deck: 4 (one King from each of the four suits)

Probability Scenario Breakdown

The question states that four cards are randomly selected. The key condition provided is that the first two of these selected cards are kings. We need to determine the probability that the third card drawn is also a king.

Initial State of the Deck

Before any cards are drawn, the deck is complete:

• Total number of cards: 52
• Number of King cards: 4

First Two Cards Drawn: Kings

Since the first two cards drawn are kings, these specific cards are removed from the deck. This is a process of sampling without replacement, meaning the deck changes after each draw.

• After the first King is drawn, the deck now has \(52 - 1 = 51\) cards remaining. The number of King cards remaining is \(4 - 1 = 3\).
• After the second King is drawn (from the deck of 51 cards), the deck further reduces to \(51 - 1 = 50\) cards. The number of King cards remaining is \(3 - 1 = 2\).

So, after the first two King cards have been drawn, the updated state of the deck for the next draw is:

• Total number of cards remaining: 50
• Number of King cards remaining: 2

Calculating Third Card Probability

Now, we need to calculate the probability that the third card drawn is a King, given the current state of the deck (50 total cards, 2 kings). The formula for probability is:

$$P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

For the third card to be a King:

• Number of favorable outcomes (remaining kings): 2
• Total number of possible outcomes (remaining cards): 50

Therefore, the probability that the third card is a King is:

$$P(\text{third card is a king}) = \frac{2}{50}$$

This calculation directly addresses the conditional probability based on the events that have already occurred.