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Find out why the formula we use to work out conditional probabilities is true!
4 x 3 = 4 + 4 + 4
4 ^ 3 = 4 x 4 x 4
4 ^^ 3 = 4 ^ 4 ^ 4
4 ^^^ 3 = 4 ^^ 4 ^^ 4
4 ^^^^ 3 = 4 ^^^ 4 ^^^ 4
They all follow the same pattern. For up arrows, you just have one less up arrow than in the original problem.
You are correct that
3 ^^ 4 = 3 ^ 3 ^ 3 ^ 3
However,
3 ^^^^ 3
is actually = 3 ^^^ (3 ^^^ 3)
Working on the parentheses first, you'd find that (3 ^^^ 3) = 3 ^^ (3 ^^ 3)
Again, working on the parentheses, (3 ^^ 3) = 3 ^ 3 ^ 3 = ~7.6 trillion. Put that back in your other problem
and you've got 3 ^^ (~ 7.6 trillion), which means you're doing 3 ^ 3 ^ 3 ^ 3... until you've done it 7.6 trillion times.
Once you get that crazy huge number, you'd put it back in the original problem:
3 ^^^ (crazy huge number).
Which is 3 ^^ 3 ^^ 3 ^^ 3... until you've done it a crazy huge number of times. Unimaginably big. And that's only with only 4 up arrows.