You and your friends are playing Amogus- a social deduction game where every now andthen, someone is voted the most “sus” and knocked out of the game. There are 7 players:you are blue, and your friends are red, green, yellow, violet, orange, and gray. Of course,no game of Amogus is complete without “that one player” who just votes for someone atrandom every round and doesn’t actually play the game properly- this time, it’s the greenplayer (I won’t name names). Suppose they start a new game, so nobody has been eliminatedyet. Before the first round of voting, orange declares, “I swear, green, if you vote for me thisround, I’ll vote for you in the next round and for the rest of the game!”Before proceeding, there are a few assumptions that need to be made:• Everyone has an equal chance of being picked by green in every round (note that nobodycan vote for themself). Also, green never abstains from voting.• Orange is being serious. If green votes orange in round 1, then orange will vote greenin round 2.• If green does NOT vote for orange in round 1, then orange will vote for someone atrandom in round 2 (what a hypocrite...). Orange never abstains either.- In that case, assume everyone has an equal chance of being picked by orange (in-cluding green, but again, orange cannot vote for themself).- In addition, orange’s round 2 vote is independent of green’s round 2 vote in everyway (both unconditionally independent and conditionally independent given green didnot vote orange in round 1).• Nobody else was going to vote for green or orange, so there is no need to conditionon who got knocked out in the first round (since who exactly was knocked out has nobearing on the probabilities below if it wasn’t orange or green).With all that established...(a) What is the (unconditional) probability that green and orange vote for eachother in the second round of voting?(b) Let’s say the first round of voting passes, and gray gets knocked out of thegame, though you weren’t paying attention to see who voted for whom. Given that inthe second round of voting, orange voted for green, what is the conditional probabilitythat green voted for orange in the first round?
You and your friends are playing Amogus- a social deduction game where every now and
then, someone is voted the most “sus” and knocked out of the game. There are 7 players:
you are blue, and your friends are red, green, yellow, violet, orange, and gray. Of course,
no game of Amogus is complete without “that one player” who just votes for someone at
random every round and doesn’t actually play the game properly- this time, it’s the green
player (I won’t name names). Suppose they start a new game, so nobody has been eliminated
yet. Before the first round of voting, orange declares, “I swear, green, if you vote for me this
round, I’ll vote for you in the next round and for the rest of the game!”
Before proceeding, there are a few assumptions that need to be made:
• Everyone has an equal chance of being picked by green in every round (note that nobody
can vote for themself). Also, green never abstains from voting.
• Orange is being serious. If green votes orange in round 1, then orange will vote green
in round 2.
• If green does NOT vote for orange in round 1, then orange will vote for someone at
random in round 2 (what a hypocrite...). Orange never abstains either.
- In that case, assume everyone has an equal chance of being picked by orange (in-
cluding green, but again, orange cannot vote for themself).
- In addition, orange’s round 2 vote is independent of green’s round 2 vote in every
way (both unconditionally independent and conditionally independent given green did
not vote orange in round 1).
• Nobody else was going to vote for green or orange, so there is no need to condition
on who got knocked out in the first round (since who exactly was knocked out has no
bearing on the probabilities below if it wasn’t orange or green).
With all that established...
(a) What is the (unconditional)
other in the second round of voting?
(b) Let’s say the first round of voting passes, and gray gets knocked out of the
game, though you weren’t paying attention to see who voted for whom. Given that in
the second round of voting, orange voted for green, what is the conditional probability
that green voted for orange in the first round?
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