With every danger, the player will either pass or fail. Suppose upon every failure the level is reset for simplicity's sake. There is a probability for each danger: For x number of times that the player tries, how many times will they pass? This probability would be shown as a fraction:

__Pass__

Attempts

For example, if the player will theoretically fail for every time they pass, it would be 1/2, or 50%. Now, why do I bring this up? This can be used to calculate the overall difficulty of a level (as compared to others).

Suppose you find the pass/attempt ratio of every danger. If you take all of these fractions and multiply them together, you will get the theoretical probability of the player passing the level. For example, a level that has a danger with a 1/2 chance of passing, a danger with a 1/3 chance of passing, and a danger with a 3/4 chance of passing. The level will have a 1/8 chance of being passed. This means you can expect your player in a perfect world to fail 7 times before passing in this particular level. In the real world, players will obviously fail more or less times than this, but this gives us an approximate figure.

Now, here is another example. Suppose this is a list of pass/attempt ratios of each danger in the level.

1/8

3/10

2/7

5/11

1/20

This would be a very, very difficult level. The chances of passing are 3/12320, meaning you can expect the average player to fail in excess of 4100 times before succeeding, and therefore should probably never ever be made. I'm just using this extreme example to make a point.

Now, what effect would a checkpoint have on this level? Let's say we stuck it in here:

1/8

3/10

2/7

=CHECKPOINT=

5/11

1/20

Now let's separate the level into two groups: pre-checkpoint and post-checkpoint. The chance of passing the pre-checkpoint portion is 3/280, or about 92 failures per pass. The chance of passing the post-checkpoint portion are 1/44. Since the post-checkpoint portion does not depend on the first portion's passing for every trial, we can simply add the number of approximate failures of each portion to get our total probability of passing the level. As a result, we can expect the player to fail ~136 times before passing, as opposed to the original ~4100.

Now, let's suppose each trial of the level took 10 seconds. 136 failures * 10 seconds = 1360 seconds, or 22m 40s. 4100 failures * 10 seconds = 41000, or 11h 23m 20s. This is a difference of 11h 40s. Note that these calculations do not take into account trials that do not reset on failure, or the fact that trials will have different lengths of time.

I hope you can see why a checkpoint would be a good idea in this case. Checkpoints aren't always a good idea, but they generally lower the frustration much more than they lower the actual difficulty.