The TV show Survivor: South Pacific had a probability problem last night. The two tribes merged, meaning that the game shifted from two separate tribes into one individual tribe. However the tribes each had six members at the merge and appeared to be headed to a tie vote. The six members of Savaii tribe would vote for one of the Upolu members to leave the game and vice versa. The problem gets a little more interesting when the threat of hidden immunity idols, challenge idols, and the tie breaking rules of Survivor are considered.
For those of you unfamiliar with Survivor terminology, a short explanation. The challenge idol prevents the bearer from having votes cast against him. They cannot be voted out of the game. The hidden immunity idol can be cast after votes are cast but before the votes are tallied. Whoever gets the hidden immunity idol played on him has all votes cast against him rendered null, protecting him from getting voted out of the game. The tie breaking rule of Survivor is to "draw rocks." In the case of a tie vote, everyone votes again only this time the only people who can be voted for are the people who tied in the vote. If there is another tie, then everyone except for those people draw rocks. Whoever draws the one black rock is eliminated from the game.
Without those considerations, the probability is straightforward to calculate. A Savaii tribe member is voted out 50% of the time and an Upolu tribe member is voted out 50% of the time. Both sides had hidden immunity idols. Played optimally, the hidden immunity idols would cancel each other out and the odds remain at 50-50. However Savaii happened to win two challenge idols and--along with the hidden immunity idol--were able to shield three of the tribe members from votes. How does this change the odds?
First let's assume both teams play the optimal strategy. This didn't happen last night. The optimal strategy isn't to decide who to vote and who to shield hours before the vote. Doing this allows for the possibility of the other tribe either figuring out who the voting target was or the possibility of a tribe member betraying this information to the other tribe. Instead the voting target and the hidden immunity shield target should be chosen randomly as close to the voting time as possible. By randomly it does not necessarily have to be equally random--you can weigh certain alternatives more heavily than others--but it does have to be random and all possibilities have to have a chance. This is the optimum mixed strategy.
Now for the odds. There are four voting scenarios. Scenario One: both tribes play their hidden immunity idols but both tribes votes for someone not protected by the idol. They draw rocks to decide and someone is randomly sent home. Scenario Two: both tribes play their hidden immunity idols and both tribes successfully block the other tribe's vote. They draw rocks to decide and someone is randomly sent home. Scenario Three: both tribes play their hidden immunity idols. Savaii successfully shields the targeted member while Upolu fails to shield the correct member. Upolu loses a member. Scenario Four: same as the last scenario only Savaii fails to shield and loses a member while Upolu successfully shields a member.
Savaii has better odds because two of its members are always protected from elimination by having won challenge idols. In Scenario One and Two, only 3 Savaii members will have to draw rocks while 5 Upolu members will have to draw rocks. Scenario Three is also more likely to happen than Scenario Four. The odds of each Scenario:
Chance of Occuring
Scenario 1 63%
Scenario 2 4%
Scenario 3 21%
Scenario 4 13%
And within each scenario, here are the chances of Savaii losing a member and Upolu losing a member.
Savaii Loss Upolu Loss
Scenario 1 23% 39%
Scenario 2 2% 3%
Scenario 3 0% 21%
Scenario 4 13% 0%
37.5% 62.5%
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