OK, so when b is high, we basically want to bet p of our bankroll. This is easy to remember. I would like to think of the Kelly bet as kind of an adjustment on that.

Given fixed b, the Kelly bet is a line from 0% at the break-even point to 100% on sure things. So where is the break-even point?

The break-even point is where the top of the formula equals zero, ie p(b+1)−1=0, so p=1b+1. I prefer to think in terms of r, the “returns”, r=b+1. (If there’s an opportunity to “double” your money, r=2 while b=1; so I think r is more how people intuitively think about things.) So the break-even is p=1r. That’s also pretty easy to remember.

So we can just check whether p is above 1r, which is a little easier than calculating the expected value of the bet to check that it’s above zero.

As noted earlier: when b is really high, we bet p, sliding from 0 to 1 as p slides from 0 to 1. We can now state the adjustment: when b is not so high, the Kelly formula adjusts things by making the slide start at p=1r instead of at zero.

So one way to calculate Kelly is to ask how much along the way between 1r and 1 we are.

For example, if r=2, then things start to be profitable after p=50%. If p=80%, we’re 3/5ths of the way, so would invest that part of our bankroll.

If r=3, then things start to be profitable after p=33.¯¯¯3%. If p=66.¯¯¯6%, then it’s half way to 100%, so we’d spend half of our bankroll.

OK, so when b is high, we basically want to bet p of our bankroll. This is easy to remember. I would like to think of the Kelly bet as kind of an adjustment on that.

Given fixed b, the Kelly bet is a line from 0% at the break-even point to 100% on sure things. So where is the break-even point?

The break-even point is where the top of the formula equals zero, ie p(b+1)−1=0, so p=1b+1. I prefer to think in terms of r, the “returns”, r=b+1. (If there’s an opportunity to “

double” your money, r=2 while b=1; so I think r is more how people intuitively think about things.) So the break-even is p=1r. That’s also pretty easy to remember.So we can just check whether p is above 1r, which is a little easier than calculating the expected value of the bet to check that it’s above zero.

As noted earlier: when b is really high, we bet p, sliding from 0 to 1 as p slides from 0 to 1. We can now state the adjustment: when b is not so high, the Kelly formula adjusts things by making the slide start at p=1r instead of at zero.

So one way to calculate Kelly is to ask how much along the way between 1r and 1 we are.

For example, if r=2, then things start to be profitable after p=50%. If p=80%, we’re 3/5ths of the way, so would invest that part of our bankroll.

If r=3, then things start to be profitable after p=33.¯¯¯3%. If p=66.¯¯¯6%, then it’s half way to 100%, so we’d spend half of our bankroll.

This seems way easier than trying to mentally calculate “expected net winnings over net winnings if you win”, even though it’s the same formula.

I find this explanation to be much easier to understand than SimonM/gjm’s