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.NO SPAM OTHERWISE I WILL REPORT YOU!
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.
Answers
Given : A motel has ten rooms , all located on the same side of a single corridor and numbered 1 to 10 in numerical order. Friends Molly and Polly have been allocated two separate rooms at the motel.
To find : the likely number of rooms between their rooms
Solution:
Moly probablty Rooms in between probability
Room 0 1 2 3 4 5 6 7 8
1 (1/10) 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9
2 (1/10) 2/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0
3 (1/10) 2/9 2/9 1/9 1/9 1/9 1/9 1/9 0 0
4 (1/10) 2/9 2/9 2/9 1/9 1/9 1/9 0 0 0
5 (1/10) 2/9 2/9 2/9 2/9 1/9 0 0 0 0
6 (1/10) 2/9 2/9 2/9 2/9 1/9 0 0 0 0
7 (1/10) 2/9 2/9 2/9 1/9 1/9 1/9 0 0 0
8 (1/10) 2/9 2/9 1/9 1/9 1/9 1/9 1/9 0 0
9 (1/10) 2/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0
10 (1/10) 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9
Probability 18/90 16/90 14/90 12/90 10/90 8/90 6/90 4/90 2/90
Likely number of rooms between their rooms
= 0 * (18/90) + 1 * (16/90) + 2(14/90) + 3(12/90) + 4(10/90) + 5(8/90) + 6(6/90) + 7(4/90) + 8(2/90)
= (1/90)(0 + 16 + 28 + 36 + 40 + 40 + 36 + 28 + 16)
= 240/90
=24/9
= 8/3
= 2.33 rooms
likely number of rooms between their rooms = 2.33
At most 1 room = 0 room or 1 room
= (18 + 16)/90
= 34/90
34/ 90 > 30/90
=> 34/90 > 1/3
=> Moly belief is right that there is a greater than 1/3 chance that at most one room will separate them
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