"What is the best bet after let's say 3 flips land on one side (heads) in your opinion?"
Do you believe the coin is fair? If so, heads always has the same chance as tails and there is no best bet, regardless of what has happened in the past. 3 heads in a row is a fairy high probability event and will occur fairly frequently, and it's easy enough to demonstrate things yourself. Sit down for a relatively short period of time and flip a coin 100 times and record the results. Better still take a bit more time and do it 2 or 3 hundred times.
"Is there a system or reasonable probability factor as to why streaks DO come to an end and why or why not this should be expected?"
There's a very clear "probability factor" why streaks end. Each time you get 3 heads in a row there's a 50% chance that the next flip will be tails and end the streak. If you do get another heads the 5th flip will have a 50% chance of landing tails, and so on. The probability that the streak will continue is halved every time you flip the coin again. For a 50/50 event like flipping a coin you can't expect the streak to come to an end with any particular flip because it's always equally likely that the flip will extend the streak. For events with other probabilities whether or not you should expect the streak to end is subjective. If the probability of getting a result that ends the streak is 2 in 3 continuing the streak won't be surprising. OTOH, if pick 3 draws the same number 2 nights in a row the 99.9% chance of drawing a different number the 3rd night means that most people would expect one of the other 999 numbers.
"Is a coin flips 1:2 probability equal to a digit from 1-9 being drawn in the same spot?"
It sounds to me like you're asking if a 1 in 2 chance is equal to a 1 in 10 chance.
"If the lottery officials do pre-tests just to make sure the machines are functioning properly and nothing else, then why can't they use plain, blank, unnumbered ping pong balls during the pre-tests?"
Because part of the test is looking for anything that might indicate that the results aren't perfectly random, and that means they need to know which balls are selected.
"The same digit hits in the same position 3 times, I'm playing for a fourth."
That at least makes some sense, because there's at least some chance that something happened 3 times in a row because the process is flawed and that digit has more than a1 in 10 chance. The problem is that what you see as the same position probably isn't. If either the machine or the ball set is different then its either a different digit or a different position.
"I don't think we've said anything different."
It sounds to me like we're saying very different things, but I'm not sure whether or not we mean different things.
"It is interesting the way we both agree that probability affects random. However, you stop short of saying probability adversely affects random. "
Probability doesn't have any effect on random. Something is random or it's not, and probability is a completely different issue. In terms of events such as flipping a coin, rolling a die, or selecting numbered (or unnumbered, FTM) balls random just means that there is nothing that favors one outcome over another. Probability is a tool that describes possible outcomes from events, whether the event random or not.
"Your use of the term "increasingly unlikely" has the same tone and meaning as random is limited or if you prefer, adversely affected."
Describing random as "limited" may just be your way of describing something I agree with, but it's a poor way to describe it. Something is either random or it's not, and nothing limits random. The only thing that happens is that each repetition increases the number of possible results, so that the series of still perfectly random events becomes 1 of an ever-increasing number. The results become increasingly unlikely but they never become impossible, so there is no limit on the results of random events.
"Do you play Pick 3?"
Nope. I've got no idea why anybody would play pick 3 if they understand probability. If you win today and keep playing once a day you can't expect to stay ahead for even a year and a half. I don't play the lottery in hopes of being statistically normal and losing money, but the low odds of pick 3 pretty much guarantee that's what will happen.
"everybody is acting like the numbers do not have any values. That they are just balls."
They are just balls. The numbers are there so that we can tell them apart, not because hey have any effect on which balls are selected. Numbers are convenient labels, but lots of other things would work just as well. If the balls had names on them maybe we wouldn't see people saying they came close when the winning pick 3 was Fred Martha Bill and they had played John Sam Susan. Of course a lot of players still see some pattern or relationship between balls labeled Fred, Ethel, Lucy and Ricky because they've been conditioned to associate those names with one another.
"If there is always an edge in either game why would lotteries have to cheat?"
They wouldn't, which is why they don't. They know that offering games with a house advantage and relying on random results and probability guarantees a gross profit that closely matches the game design.