A motel has ten rooms, all located on the same side of a single corridor and numbered 1 to 10 in numerical order. The motel always randomly allocates rooms to its guests. There are no other guests besides those mentioned.
a) Friends Molly and Polly have been allocated two separate rooms at the motel. What is the likely number of rooms between their rooms?
b) Molly believes there is a greater than 1/3 chance that at most one room will separate them, but Polly disagrees. Who is right? Explain why.
c) On another occasion, Molly, Polly and a third friend, Ollie, were allocated three separate rooms. Molly believes there is a better than 1/3 chance that they are all within a block of five consecutive rooms. Ollie believes that there is exactly 1/3 chance and Polly believes there is less than 1/3 chance. Who is right? Explain why.
d) Ollie arrived after rooms were allocated to Molly and Polly. There was then a 50% chance he would be in a room adjacent to Molly or Polly or both. In how many ways could a pair of rooms have been allocated to Molly and Polly?
c
d
a
a
a and b
a) To determine the likely number of rooms between Molly and Polly's rooms, we need to consider all possible configurations of room allocations. Since there are 10 rooms, we can assume that any room from 1 to 10 is equally likely to be allocated to either of them.
There are two extreme cases:
1. If Molly gets room 1 and Polly gets room 10, there would be 8 rooms between them (rooms 2-9).
2. If Molly gets room 10 and Polly gets room 1, there would be 8 rooms between them (rooms 2-9).
In the remaining cases, the number of rooms between Molly and Polly's rooms would be any value from 0 to 7.
Therefore, the likely number of rooms between Molly and Polly's rooms would be in the range of 0 to 8.
b) Molly and Polly have a disagreement about the likelihood of at most one room separating their rooms. To settle this, we need to calculate the probability of this event occurring.
Let's consider the two extreme cases discussed in part (a) above, where Molly and Polly get rooms 1 and 10, respectively. In this case, there is no room separating them.
In all other cases, there would be at least one room separating Molly and Polly's rooms. This includes scenarios such as:
- Molly in room 1 and Polly in any room from 2-9.
- Molly in any room from 2-9 and Polly in room 10.
Since there are 90 possible room allocations (10 choices for Molly's room multiplied by 9 choices for Polly's room), only 2 out of the 90 cases (the extreme cases) do not have any rooms between them.
Therefore, the probability of at most one room separating Molly and Polly's rooms is 2/90, which simplifies to 1/45.
Based on the calculation, Molly is right, and the probability is less than 1/3.
c) To determine who is right about the probability of Molly, Polly, and Ollie being allocated rooms within a block of five consecutive rooms, we need to consider the different room allocations.
Since each friend can be allocated any of the ten rooms, there are a total of 10^3 = 1,000 possible room allocations.
Now, let's consider the cases where all three friends are allocated rooms within a block of five consecutive rooms:
1. Molly, Polly, and Ollie are allocated rooms 2, 3, and 4.
2. Molly, Polly, and Ollie are allocated rooms 3, 4, and 5.
3. Molly, Polly, and Ollie are allocated rooms 4, 5, and 6.
4. Molly, Polly, and Ollie are allocated rooms 5, 6, and 7.
5. Molly, Polly, and Ollie are allocated rooms 6, 7, and 8.
6. Molly, Polly, and Ollie are allocated rooms 7, 8, and 9.
7. Molly, Polly, and Ollie are allocated rooms 8, 9, and 10.
In each of these cases, all three friends have their rooms within a block of five consecutive rooms. Therefore, there are 7 such room allocations.
Hence, the probability of all three friends being allocated rooms within a block of five consecutive rooms is 7/1,000, which is less than 1/3.
Based on the calculation, Polly is right, and the probability is less than 1/3.
d) To determine the number of ways Molly and Polly's rooms could have been allocated, given that there was a 50% chance that Ollie's room would be adjacent to Molly's or Polly's or both, we need to consider the different possibilities for Ollie's room.
Since Ollie has arrived after Molly and Polly's rooms were allocated, he has three possible cases:
1. Ollie's room is adjacent to Molly's room.
2. Ollie's room is adjacent to Polly's room.
3. Ollie's room is adjacent to both Molly's and Polly's rooms.
Let's calculate the number of ways for each case:
1. Ollie's room adjacent to Molly's: In this case, Molly and Ollie each have 8 choices (rooms 2-9), and Polly has 9 choices (rooms 1 and 3-10). Therefore, there are 8 * 8 * 9 = 576 possible allocations.
2. Ollie's room adjacent to Polly's: Similar to the previous case, Molly and Ollie each have 8 choices, and Polly has 9 choices. So, there are also 576 possible allocations.
3. Ollie's room adjacent to both Molly's and Polly's: In this case, Molly, Ollie, and Polly each have 8 choices, resulting in 8 * 8 * 8 = 512 possible allocations.
Adding up the possibilities for each case, we get a total of 576 + 576 + 512 = 1,664 ways to allocate rooms to Molly and Polly.
Therefore, there are 1,664 ways a pair of rooms could have been allocated to Molly and Polly, given the conditions.