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?
2. Find the probability of the event that the 13th card dealt is the first King dealt.
1/13
2. 1/13*4*(18 12)/(52 13)
1. To find the probability that the 13th card dealt is a King, we need to determine the number of favorable outcomes and the total number of possible outcomes.
The number of favorable outcomes: There are a total of 4 kings in a standard 52-card deck.
The total number of possible outcomes: When dealing the 13th card, there are 52 cards remaining in the deck.
Therefore, the probability that the 13th card dealt is a King is 4/52, which simplifies to 1/13.
2. To find the probability that the 13th card dealt is the first King dealt, we need to consider the order in which the cards are dealt.
The number of favorable outcomes: There is only one possible outcome where the 13th card dealt is the first King dealt.
The total number of possible outcomes: To determine the total number of outcomes, we need to consider the number of ways the remaining 12 cards can be arranged. Since the first King can be dealt anywhere between the first card and the 13th card, there are 12 possible positions for the first King. The remaining 12 cards can be arranged in (52-1)!/(52-13)! = 12! ways.
Therefore, the probability that the 13th card dealt is the first King dealt is 1/(12!), which is approximately 0.0000000053 or 1 in 479,001,600.
To find the probability of certain events in this scenario, we can use basic counting principles and fractions.
1. To find the probability that the 13th card dealt is a King, we need to determine the number of favorable outcomes (getting a King on the 13th card) and the total number of possible outcomes (all possible sequences of 13 cards).
The total number of possible outcomes is the number of ways to arrange 13 cards from a standard 52-card deck, which is given by the binomial coefficient: C(52, 13) = 52! / (13! * (52-13)!) = 635,013,559,600.
Now, the number of favorable outcomes is the number of ways to choose a King on the 13th card, which is 4 (since there are 4 Kings in a deck).
Therefore, the probability of the 13th card dealt being a King is 4 / 635,013,559,600 ≈ 6.31 × 10^-12.
2. To find the probability of the event that the 13th card dealt is the first King dealt, we need to consider the arrangement of the cards in the sequence.
The first King can be any of the 13 cards, and the remaining 12 cards can be any combination of the remaining 51 cards that are not Kings.
The number of favorable outcomes is then 13 * (51! / (12! * (51-12)!)) = 13 * C(51, 12) = 78,884,285,600.
The total number of possible outcomes remains the same as in the previous question: 635,013,559,600.
Therefore, the probability of the 13th card dealt being the first King dealt is 78,884,285,600 / 635,013,559,600 ≈ 0.124 or 12.4%.