I'm just going to express again that I hate luck tasks, especially with Bates and Anthony checking in 2nd after like searching 2 sandcastles. I don't blame the task for Matt and Daniel's elimination, since they were too arrogant with the boat task.
I'm shocked they all took the penalty. If I was racing, I would have not agreed to take the penalty hoping that I have kept good track at which sandcastles I knocked down, but then again, I wasn't out there for 3 hours either.
Ain't nobody got time for a 4-hour penalty when almost half of the sandcastles are already knocked down. Even if you didn't take the penalty, it will still take less time to knock down and rebuild them. Forget the pact, I'm worrying about myself.
Let's look at the mathematics of this. There are 400 sandcastles at the start and 10 clues; the first team has a 2.5% chance each try. It appeared to be slow going and I thought the average time per sandcastle was running at about 3 minutes. That means it would take on average 1 hour of attempts to find a clue. For each succeeding team, the odds went down until for the final 3 teams there are 3 sandcastles left, for a probability of 0.75% for each try. That means is would take 3.3 hours for them to find a clue. They appeared to have gone beyond that, so the statistics of the situation (which had worked for all prior teams) did not appear to be working out for the last 3 teams. Their election of taking the 4 hour penalty together meant that one team would almost certainly be eliminated and the other two would be fighting against elimination at the end of leg 2.
Wasn't there 11 clues at the start?
If there were only 10 clues, somebody was guaranteed not to get one and thus forced to take a penalty. Going with that new addition, the first team had a 2.75% chance of getting the clue at each sandcastle. Going with the assumption that each sandcastle would take on average 3 minutes to knock down and rebuild, there would be a 55% chance that a team would get it within the first hour of searching. Obviously this will decrease as teams find clues, but at the same time, I think from watching it, no teams rebuilt the last sandcastle they knocked over (the one with the clue), so going with that assumption...
10 teams: 50.12% chance within one hour
9 teams: 45.22% chance within one hour
8 teams: 40.3% chance within one hour
7 teams: 35.35% chance within one hour
6 teams: 30.38% chance within one hour
5 teams: 25.38% chance within one hour
4 teams: 20.36% chance within one hour
3 teams: 15.31% chance within one hour
This is based also on another assumption that the teams are doing it independently of one another (very poor statistical assumption but is basic enough to see the effects of finding the clues are). Thus, if Katie, Jennifer and Daniel had continued with the Roadblock, they would have taken about 6 and a half hours to complete (with all the assumptions used)!
However, going with the assumption that the Roadblockers DID
rebuild the last sandcastle, the effects of chance change significantly:
10 teams: 50% chance within one hour
9 teams: 45% chance within one hour
8 teams: 40% chance within one hour
7 teams: 35% chance within one hour
6 teams: 30% chance within one hour
5 teams: 25% chance within one hour
4 teams: 20% chance within one hour
3 teams: 15% chance within one hour
Although insignificant in some respect, if the last sandcastle was to be built back up again, the chances of find the sandcastle would be reduced. Once again, these calculations are done with the assumption that all the teams performances are independent of each other. Thus, the time taken for the three teams to find a clue at 3 teams left would be slightly greater at 6 hours 40 minutes.
However, these models are really simplistic, ignoring a number of factors including:
- How quickly teams are constructing the sandcastles
- How many sandcastles teams know have been searched
- Teams potentially "missing" the clue (remember how Dave said he might have missed a clue for not digging deep enough?)
- How many sandcastles other teams deconstruct before finding a clue
There are other factors but I think these could be considered as the most important. At the end of the day, looking over the calculations here would suggest that taking the penalty would have been a good decision, however, incorporating the factors affecting how long the teams take, it may not have been such a wise decision after all. I guess time will tell...