Rolling Five Fair Dice

A middle school teacher emailed me a really interesting problem earlier today:

I have a probability question for you that I hope you can answer. What is the probability of rolling 5 dice and getting 66553 in 3 attempts? It's not for any class or anything, a friend of mine wanted to know. (Apparently this is some bar room game of chance).

The situation reminded me of the game "Liar's Dice" featured in the movie Pirates of the Caribbean: Dead Man's Chest.

In the game, each player has five dice in a cup. I don't know whether Liar's Dice is the game the teacher references, but that's the first thing that came to mind. On to the problem...

Each of the five dice can show one of the values from {1, 2, 3, 4, 5, 6}. We can think about all five dice being different colors. That makes the outcome 66553 different than 65653. Thinking this way will help us count all possible outcomes.

I have a PDF of the solution I wrote up to this problem below. I plan to use this problem the next time I review independent events, dependent events, and mutually exclusive events in stats class.

Ways to Win the Dice Game Solution

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See the PDF for the continuation of the solution I wrote. Please feel free to comment below.

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