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# Risk game

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You're playing a variant of the board game Risk with your friends. In this variant, the attacker gets to roll two 7-sided dice and the defender rolls a single 8-sided die. The attacker wins if at least one of their dice is strictly higher than the defender's. Find the probability that the attacker wins.

##### Solution Summary

Every step has been shown. A shortcut calculation is also explained in the solution.

##### Solution Preview

Step 1) Total number of ways to roll the dice = 7 * 7 * 8 = 392

Step 2) Calculate the probability the defender wins in each case, and use the complement later to find the probability the attacker wins. Note that the game has no ties.

Case 1) Defender rolls 1:
Defender wins if the attacker rolls
(1, 1)

Case 2) Defender rolls 2:
Defender wins if attacker rolls
(1, 1), (1, 2)
(2, 1), (2, 2)

Case 3) Defender rolls 3:
defender wins if attacker rolls
(1, 1), (1, 2), (1, 3)
(2, 1), (2, 2), (2, 3)
(3, 1), (3, 2), (3, 3)

Case 4) Defender rolls 4:
defender wins if attacker rolls
(1, 1), (1, 2), (1, 3), (1, 4)
(2, 1), (2, 2), ...

Solution provided by:
###### Education
• MSc, California State Polytechnic University, Pomona
• MBA, University of California, Riverside
• BSc, California State Polytechnic University, Pomona
• BSc, California State Polytechnic University, Pomona
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• "Hello, thank you for your answer for my probability question. However, I think you interpreted the second and third question differently than was meant, as the assumption still stands that a person still independently ranks the n options first. The probability I am after is the probability that this independently determined ranking then is equal to one of the p fixed rankings. Similarly for the third question, where the x people choose their ranking independently, and then I want the probability that for x people this is equal to one particular ranking. I was wondering if you could help me with this. "

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