A player is randomly dealt a sequence of 13 cards from a standard 52-card deck. All sequences of 13 cards are equally likely. In an equivalent model, the cards are chosen and dealt one at a time. When choosing a card, the dealer is equally likely to pick any of the cards that remain in the deck.

1. What is the probability the 13th card dealt is a King? Note: Your answer should be a number. Do not enter '!' or combinations in your answer.

2. Find the probability of the event that the 13th card dealt is the first King dealt. Identify the correct expression.

The probability that a given rank of card is chosen from the remainder of the deck (matching suit not specified) 4/(52-nth card)) multiplied by each previous card in the set.

1.6971 x 10^-14

1/13 * 4(48 12)/(52 13)

45

1. The probability that the 13th card dealt is a King can be calculated by dividing the number of ways to choose a King for the 13th card by the total number of possible sequences of 13 cards.

There are 4 Kings in a standard 52-card deck, so the number of ways to choose a King for the 13th card is 4. The total number of possible sequences of 13 cards is the same as the number of permutations of 52 cards taken 13 at a time, which is denoted as "52P13." This can be calculated as:

52P13 = 52! / (52 - 13)! = 52! / 39!

Now, we can calculate the probability:

Probability = (number of desired outcomes) / (total number of possible outcomes)
= 4 / (52P13)

Therefore, the probability that the 13th card dealt is a King is:

Probability = 4 / (52P13)

2. The probability of the event that the 13th card dealt is the first King dealt can be determined by dividing the number of ways to choose a sequence of 12 non-King cards followed by a King for the 13th card, by the total number of possible sequences of 13 cards.

The number of ways to choose a sequence of 12 non-King cards is equal to the number of permutations of 48 non-King cards taken 12 at a time, denoted as "48P12." This can be calculated as:

48P12 = 48! / (48 - 12)! = 48! / 36!

The number of ways to choose a King for the 13th card is 4.

The total number of possible sequences of 13 cards is the same as the number of permutations of 52 cards taken 13 at a time, denoted as "52P13."

Therefore, the probability of the event that the 13th card dealt is the first King dealt can be calculated as:

Probability = (number of desired outcomes) / (total number of possible outcomes)
= (48P12 * 4) / (52P13)

The correct expression for the probability of the event that the 13th card dealt is the first King dealt is:

Probability = (48P12 * 4) / (52P13)

To answer the first question, we need to find the probability that the 13th card dealt is a King.

Since there are 52 cards in a standard deck, and 4 Kings in that deck, the probability of any randomly chosen card being a King is 4/52, which simplifies to 1/13.

Therefore, the probability the 13th card dealt is a King is also 1/13.

To answer the second question, we need to find the probability of the event that the 13th card dealt is the first King dealt.

In this scenario, we are considering the order in which the Kings are dealt.

There are 4 Kings in the deck, so there are 4 possible ways the first King can be dealt out of the 13 cards.

Since there are 52 cards in the deck, after the first King is dealt, there will be 51 cards remaining, out of which 3 are kings.

Therefore, the probability of the event that the 13th card dealt is the first King dealt can be expressed as:

(4/52) * (3/51) * (50/50) * (49/49) * ... * (2/40) * (1/39)

or more generally:

(4/52) * (3/51) * (50/50) * (49/49) * ... * ((14 - 2)/(52 - 14)) * ((13 - 1)/(52 - 13))

Simplifying this expression will give you the final probability.

question #1. Easy!

4/52