Odd of removing Saw head trap

As noted above there's a pool of, iirc, 12 searches divided randomly among the Survivors to remove a trap. This means that if everyone is trapped and everyone gets their traps off, a total of 12 box searches will have been completed.

This divides randomly per Survivor, but at no time will you ever need to search more than 4 boxes to get a trap off. If these searches are divided evenly every Survivor would pull their trap off after it third search. Variations mean two Survivors need 4 attempts, while another would need 3, and the last gets it off first try. Or only one person needs 4, two get it in 3, and the last gets it on 2. And so on...

These numbers are adjusted when using addons that change the number of traps available, but I don't know how myself. (If someone failed to get a trap off at 4 searches like OP suggested, could this have been the reason?)

Also, to combat the Pig camping a single box strat, these odds apply to all boxes, so it doesn't matter which boxes you search. Once you reach your number of searches you'll remove your trap whichever box you're on. There are also 5 boxes be default now, instead of 4.

In short, in an average Pig match, don't worry about looking for specific boxes. There's no "your box". You will get the trap of within 1-4 searches. If possible (such as playing with friends) you can keep track of how long it takes you or others to get your traps off to have an idea of what to expect. If you get it off first try, for example, your teammates are going to have a rough time of it.