probability of being under ev every month for 24 straight?

These programs that graph "EV" are probably only of the worst things to happen to poker.

surely its a 50/50 chance of being either side of ev every month?

If you're talking cumulative EV then no, because if you run a little +EV this month then you can still be -EV cumulatively.

If you're talking strictly this month's EV then it still isn't necessarily 50/50. If I bet on rolling a 7 with two dice, and am offered 6:1 odds then that's a +EV bet, but I'm more likely to run below EV than above since I'm more likely to lose a roll. So how likely you are to run below EV on a given month (as measured by the software) depends on your equities in the measured situations.

Check each month independently. You are looking at cumulative.

As alluded to above, if you finish 1 month below EV then your future expectation is to stay that much below EV forever. The reason is because your future expectation is always zero EV and past events don't affect that.

Yes, you have a 50% chance each month to fluctuate over or under 0 for that month. But you don't expect to make up past variance.

I think one factor is also how many hands you play per month.
If you only play 1 hand per month, 22 vs AA preflop all-in, it is far more likely you will be -EV, than if you play 10k hands/month.

also...
If everytime you run above EV, you increase the stakes you play and when running bad decrease your stakes/stop playing, it will be more likely you'll end up below EV in the end of the month.

Both of these statements are false. There is nothing the player can do to increase or decrease the chance of being below his all-in EV line. Number of hands is irrelevant, and style of play is irrelevant. All-in EV simply measures the luck of the deal after all player decisions have been made, and player decisions have no effect whatsoever.

You also are using the Gambler's Fallacy in your second statement, implying that because you ran bad on some hands that you are also "running bad" going forward. There is no relationship of past deals to future ones. "Running bad" is entirely a hindsight description.

As Rusty implied, this graph is the most misused and misunderstood aspect of poker tracking programs. It has no utility whatsoever for the player, and it's better to focus on the things that you can control.

No utility whatsover. You heard it here first, folks.

I don't think I entirely agree with this. For *very* small numbers of hands, if you are behind then you are more likely to run behind EV.

Consider 1 hand where you're an 80:20 dog. 80% of the time, your outcome will be -1 unit, but your expected outcome is -.6 units. So 80% of the time you'll run below EV (and 20% you'll run above EV)

At 2 hands, you have
.8*.8 = .64 to lose both (under EV)
.2*.2 = .04 to win both (above EV)
and the rest (.32) to break even (above EV)
So after 2 hands, you still expect to be under EV

I am not that inclined to find out how many hands it would be until you would expect this effect to dissipate but I imagine the answer is "not very many"

Obviously we need enough hands to begin to actually approach the EV. But in essence it appears you are saying, yes the player can affect his EV line by just stopping after a few hands. Or, only go all-in when you are sure you are ahead for your first few lifetime hands until you have a real average. Yes?

Obviously if we flip one coin we will either be way over or way under EV because we have no average. I'm not sure that really contradicts my statements.

Well, "number of hands is irrelevant" is false by my standards of nittiness. But true for practical purposes.

I stand corrected. So let's see how much difference it does make. We can do this fairly easily using the binomial distribution and a spreadsheet. Took about 5 minutes.

If strategy is to try to get it in with a dog by shoving only our worst hands, trying to be under EV, and say we are 100% correct on being the dog every time, then:

After 1 hand we have 80% chance to be under EV.
After 2 hands we have 64% chance to be under EV.
After 7 hands we have less than 58% chance to be under EV from that point on.
After 61 hands we have less than 55% chance to be under EV from that point on.
After 96 hands we have less than 54% chance to be under EV from that point on.
And obviously this continues to approach 50% as we go up.

And the above percentages can be multiplied by our certainty of shoving as the dog too. Right now they are multiplied by 100%. If our strategy were to try to always shove as the best hand and to be over EV (same pattern applies), that certainty would go down and our influence on trying to be over EV would drop even quicker, to irrelevance after 3 hands basically.

I'm actually surprised you maintain a significant underdog status that far out.