I agree viewing this as a math problem would be more accurate. Where did you get "-33 delta" and conclude the one standard deviation? I should probably be using the same website(s) ...

From my broker. Look at any quote on an option, and pull up the greeks. I use TOS web portal. I have the "Bid", "Ask" and "Delta" columns open. You can also put the "Prob ITM" column up, which is a closer approximation to the odds it will be ITM than the delta, but for quick math sake the delta works good enough (now the delta opn the 9.5 put is -34, and probability is 42.83%). Technically a 1 std move would be the 9 put, -26 delta, 33.8% probability, for $0.41, but you get the idea.

Are we talking about a standard deviation where 2/3rds of values are in the middle? In that scenario, the 1/6th of values below the middle hurt me, while the top 1/6th are still profitable. Wouldn't I lose 1/6th of the time?

(With two standard deviations, the 5% outside the range is also half favorable and half against me).

If you sell a 1 std put, your looking at where the market indicates, based on it's implied volatility, the stock is likely to move within the time left of the option. In this instance, the market believes the IV is 76.83%, and there is a 68% chance the stock will move up or down $1.13 (roughly) in the next 38 days. There is a 32% chance that it will increase or decrease more than $1.13. A 16% chance (roughly) that it will decrease more than $1.13, thus putting your option in the money. But if you sell it for $0.40, you need it to move $1.53 to lose money. Which means it's less than 16% (I don't know the actual odds off the top of my head, but roughly 7% ish).

Now that's what the market believes could happen. Actual volatility is the unknown. Maybe the stock will be more volatile, and move up or down more than $1.13. Maybe less. That's where your speculation will come in.

To get in the weeds, that's not really accurate. A standard bell curve assumes equal distance on both tails, i.e. a stock has an equal probability of increasing as it does decreasing. In reality, the probability of option pricing standard deviations doesn't follow a standard bell curve. It suffers from kurtosis (there is a greater chance of 1 std moves), and skew. A 10 std move on a standard bell curve might put M at +/- $12, lets say. It can certainly go up $12, but it can't go negative, which means it can't move down $12. So the downside is capped, pushing the bell curve to the left, and increasing the right tail. But that's getting a little too specific for our discussion.

But that's really only viewing it in a fixed period of time. As in, if you sold the option today, closed your eyes, then opened them on January 29th, what are the odds today that on the last day something will happen. But the stock prices change, and the odds change with it. If the stock moves up $0.50 tomorrow, but the implied volatility stays the same, are you more or less likely that the stock will then drop $1.63 over the next 37 days? Probably less. If volatility drops, it's less likely to make a $1.13 move in 38 days, let alone in the time left in the option.

As a general rule of thumb, the option's delta is it's percentage chance it will expire in the money. So a 30 delta has a 30% chance of expiring in the money, and a 70% chance of expiring worthless. Double the delta, and you have the chance that the option will be within the money at some point during the option's life. So a 30 delta has a 60% chance that it will be in the money at some point in the next 38 days. Not accurate numbers, but good rough approximations.

Few people hold options to expiration though. As it isn't worth it. Most people will wait for an option they sold to decrease in value, then roll it out. If you sold the 38 day $0.40 option, and it dropped to $0.20 in a week, you just made 50% of your max profit in 7 days. Is it worth it to wait another 31 days for the other 50%? Probably not. Close it now, and sell another one further out. If the option goes in the money in 7 days, do you have a losing position? Not necessarily, as you knew there was a 60% chance it could do that, but you still have 31 days left.

I'm also ignorant of how often PUT options are exercised. I assume a PUT option that is barely in the money would rarely be exercised, except on it's expiration date. But I don't know the relationship between a PUT option being in the money, and the percentage chance it gets exercised.

There are two types of options: European style, and American style. European options can only be exercised at the expiration, not early. American can be exercised at any time.

Generally, the further in the money an option is, and the closer it is to expiration, the more likely it will be exercised. Why? ITM and short term options have little extrinsic value. If you exercise an option, you lose the extrinsic value. But you could gain in something else, say a dividend.

Usually, the only reason an option will be exercised early, if american, is if there is a financial gain to do so. ~98% of the time, it's because of an upcoming dividend. An investor can exercise an option, hold the stock for the exdate, collect the dividend, then ditch the stock. Options get exercised around its exdate frequently, especially ITM options. Less likely for OTM (although not impossible).

Say the extrinsic value on the option is $0.50, but the projected dividend is $0.55. You can exercise the option (lose $0.50), hold it for a day (gain $0.55), then sell the stock ($0.05 gain). It becomes much more likely the closer to the exdate, and the closer to expiration, and the further ITM the option goes (as all have lower extrinsic value).

Some times they get exercised because someone exercised their option (usually because of an upcoming dividend). When you sell an option to someone, they likely had a spread themselves (meaning they sold an option to someone else, and bought an option from you). If the other party of their transaction exercises an option, they will exercise your option to transfer the stock. Of course, it doesn't really work this way, as option exercise is assigned randomly, not to the person you originally sold the option to, but you get the idea.

The remaining 1%? Someone makes a mistake, or just wants the stock (or get rid of the stock). Tax reasons, personal reasons, whatever. They just want to move the stock. It's rare though.