# TessHuelskamp/riddler-538

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# Descrption

Problem description from the website:

Each of the seven dwarfs sleeps in his own bed in a shared dormitory. Every night, they retire to bed one at a time, always in the same sequential order, with the youngest dwarf retiring first and the oldest retiring last. On a particular evening, the youngest dwarf is in a jolly mood. He decides not to go to his own bed but rather to choose one at random from among the other six beds. As each of the other dwarfs retires, he chooses his own bed if it is not occupied, and otherwise chooses another unoccupied bed at random.

From that we then need to find:

1. The probability that the oldest dwarf sleeps in his own bed?
2. The expected number of dwarfs who do not sleep in their own beds?

# Solution Explanation

NB: Instead of listing dwarves from oldest to youngest in terms of numbers (0-6, e.g.), I decided to name the dwarfs A-G with A being the youngest and G being the oldest. This made the most sense to me and was easy to spot check that things were going well.

I started by trying to list all of the decsions the dwarves could make by hand. That strategy worked well when dwarf A happened to choose bed G (for that round, dwarves B-F would choose their own bed and dwarf G would be left with dwarf A's bed), but didn't when the dwarves happened to make decisions that would affect other swarves' decisions.

Instead, starting with the 6 choices dwarf A could make, I decided to code up all of the possible choices the dwarves could make and the likelhood that they would happen. So, dwarf A could make 6 choices at the beginning (beds B-G) and each of those choices have a 1/6.0 chance in occuring. Then, when B goes to choose a bed, if B's bed isn't taken, B will take his own bed. If it isn't however, B has an additional 6 choices to make (beds A or C-G). The 6 choices B could make each have a 1/36 (1/6 from the choice dwarf A made times 1/6 from the choices B can make). I store those arrangements and the probability they happen and then move onto placing dwarves C-G in each of the arragements from the previous step.

At the end, I calcuate the results using how likely each of the arrangements are and where each dwarf is placed in that particular arrangement.

# Results

We were trying to find:

1. The probability that the oldest dwarf sleeps in his own bed?
• 0.416
2. The expected number of dwarfs who do not sleep in their own beds?
• 2.858

Also here's the percentage that each dwarf is in their own bed:

• A 0
• B 0.833
• C 0.805
• D 0.766
• E 0.708
• F 0.611
• G 0.416

# Update

This solution was the random winner! Link