Laura and Adrian attend a dinner party. The 12 guests are seated at random
around a round table, with all seating arrangements equally likely.
(a) What is the probability that Laura and Adrian sit next to each other.
(b) Explain your answer by giving a clear description of an equally-likely-
outcomes model on which it is based. In other words, tell me what
you are using, how many elements it has, and how this implies your answer from
part (a).
why is the answer 2/11 in part a and not 2/12? Thanks
One of the problems with the "round table" seating is in the interpretation.
The most common one is that if everybody stood up and moved one seat to the right, the seating arrangement is still the same.
So consider laura and Adrian seated. Then there are 10! ways to sit everybody else going clockwise and 10! counterclockwise. Which would be 2*10! ways.
But Laura and Adrian could also be switched
My answer : 2*2*10! or 4*10! ways to seat with Laura and Adrian together
without restrictions, there would be 2*11! ways , sit the first person, then the other 11 either clockwise or counter-clockwise
prob of the event = 4*10!/2*11! = 2/11
The answer is 2/11 and not 2/12 because in a round table with 12 guests, there are only 11 possible adjacent seating arrangements.
Let's break down the solution step by step to understand why the probability is 2/11:
(a) To find the probability that Laura and Adrian sit next to each other, we need to determine the total number of equally likely outcomes where they are seated next to each other.
To do this, we can consider placing Laura and Adrian as a single entity. This reduces the problem to arranging 11 people around a circular table. Since we treat Laura and Adrian as a single entity, we now have 11 elements to arrange.
Now, out of these 11 elements, the number of ways in which Laura and Adrian can sit next to each other is 2. They can either sit in the order "Laura-Adrian" or "Adrian-Laura".
Therefore, the probability of Laura and Adrian sitting next to each other is 2/11.
(b) The equally-likely-outcomes model in this case is based on the assumption that all seating arrangements are equally likely. This means that there are no restrictions or preferences for any specific arrangement.
When considering the circular table with 12 guests, there are a total of 12 elements to arrange. However, since Laura and Adrian must sit next to each other, we treat them as a single entity, reducing the number of elements to 11.
If we ignore the circular nature of the table, we would have 11 seats available for them to sit next to each other out of the 12 possible seats. However, since the table is round, we need to account for the fact that the first and last seat are adjacent as well. This means we only have 11 possible adjacent seating arrangements.
Hence, the probability is 2/11, not 2/12.
To calculate the probability that Laura and Adrian sit next to each other, we need to determine the total number of equally likely outcomes and the number of favorable outcomes where Laura and Adrian sit next to each other.
(a) To calculate the probability, we need to determine the number of favorable outcomes and the total number of outcomes.
The total number of outcomes can be calculated using the concept of circular permutations. For a round table with 12 seats, there are (12-1)! = 11! ways to arrange the 12 people without considering the round table restriction (treating it as a line). This gives us a total of 11! possible arrangements.
Now, let's consider the favorable outcomes where Laura and Adrian sit next to each other. We can treat Laura and Adrian as a single entity, which means there are 11 entities (including the pair of Laura and Adrian) to be arranged around the table. This can be done in (11-1)! = 10! ways. Additionally, within this arrangement, Laura and Adrian can be arranged as 2! = 2 ways.
Therefore, the total number of favorable outcomes is 10! * 2!.
The probability that Laura and Adrian sit next to each other is given by the number of favorable outcomes divided by the total number of outcomes:
P(Laura and Adrian sit next to each other) = (Number of favorable outcomes) / (Total number of outcomes)
= (10! * 2!) / 11!
Simplifying this expression, we get:
= (10! * 2) / 11*10!
The factor of 11 in the denominator cancels out with the 11 in the numerator, leaving us with:
= 2 / 11
Therefore, the probability that Laura and Adrian sit next to each other is 2/11.
(b) To explain the answer, we used an equally-likely-outcomes model based on circular permutations. The model assumes that all seating arrangements are equally likely. We considered the total number of outcomes (11!) where the 12 guests are seated randomly around the table. We then calculated the number of favorable outcomes, which is the number of ways Laura and Adrian can sit next to each other (10! * 2!). Dividing the number of favorable outcomes by the total number of outcomes gives us the probability of Laura and Adrian sitting next to each other, which is 2/11.