Question

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.

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.

Answer 1

Ten rooms are located on the single side if the corridor and numbered 1 to 10 in numerical order. The room allocation is random.

Molly, Polly and a third friend Ollie were allocated three seperate rooms.

To find the chance that they are all within a block of five consecutive rooms.

Now, 10 rooms can be allocated to 3, in 10P3 number of ways, ie. 10*9*8 number of ways; as there are 10 room options for the first person, 9 for the second person and 8 for the third person.

So, all possible cases

=10*9*8

=720.

Now, in 10 rooms, there are 6 combinations of 5 consecutive room blocks. These are {1,2,3,4,5}, {2,3,4,5,6}, {3,4,5,6,7}, {4,5,6,7,8}, {5,6,7,8,9}, and {6,7,8,9,10}.

So, this block of 5 rooms can be selected in 6 ways.

For each of them, 3 rooms can be allocated from 5 in 5P3, ie 5*4*3 number of ways; as there are 5 room options for the first person, 4 and 3 for the second and third person respectively.

So, favourable cases

=6*5*4*3

=360

So, the required probability is

=360/720

=1/2.

So, the probability is 1/2.

Now, 1/2 is greater than 1/3, so among the three friends, Molly is right.