Balancing in push your luck games

Look at it like this:

A chance to win/loose is 50%.
The next step also has this chance, 50%.

The total chance of winning both steps is only 25%. Players consider this to be very low, to go all the way with push your luck.

You need a higher chance for winning. Or multiple steps for the player to walk on while the chances are high.

***

You need a percentage per batch, and see if the end percentage is somewhere around what you want. If it is 50% per player, for the whole game combined. You need to look at the number of push your luck moments.

If this is 3. Simply third root the 0,5. This will end up with 80% chance of winning per push your luck.
Know that the bust is 20% for each moment. But if players keep trying to push their luck. The total game will have a 50% again.
20% + 0,8*20% + 0,64*20% = 49%

Now, I don't know how many moments you have that count as push your luck moments.
I also don't know what your end goal winning percentage should be for all moments to be successes.
Nor do I know if you want each moment to be equal. Or that you want something along the lines of 90%*80%*70% (=50%).

But I do know that if you have push your luck moments and players need to compete with these against each other. The game needs something to encourage players to take the gamble if they feel they fall behind.

What is the penalty? Does the player loose 50% or so?
What does the player win? The best balances that I know, ask for an equal win reward as what the player may loose.
+50% or -50%
+100% or -100%
There is no +100% and -50%. Unless you want players to make a steady progress. But then, they will always take the gamble.

If you have questions, please ask.
If this post of mine was a complete crap in your eye's, tell me too. :)

Don't worry about not knowing math. You guessed perfectly right on the number of cards that a player will pick.

The moment of push your luck. Well, you have described your game as one big moment. That is ok.

There are games out there that require the player to plan ahead their push of luck by having multiple smaller moments with great risk. Although the chance to loose per moment is smaller. The total chance of losing it all would be considerate bigger.

War games are a very fine example of this.

***

Regarding what you have described.

If white cards are safe, and you are allowed to pick one other card with a colour. The number of cards that a player can pick while taking no risk at all.
1: 100%
2: 50%
3: 25%
4: 12,5%
5: 6,25%

While chances are halved each time. It is better to look at the average number of cards that a player will be drawing in a long run. It is 2 on average for an infinite deck. 1-2 cards would be if you only consider the result on the first draw.

A little warning, I have not yet looked at having either a blue or red card. It will be a little higher then the average of 2.

Because I think that psychologically, players will stop drawing cards, once they have drawn either a blue or red card. The chances are just too high to bust. Especially with what you have gained so far.

If the risk is 25% to bust. Players could risk this at a maximum of 3 cards. Just to gain a 4th. But having 4 or more cards. And having that chance to bust. Well, the reward is not worth the risk.

The bust will be 25% chance to loose 100%. That is -25% in math terms. The reward needs to be bigger on average for an attempt.

The reward on 2 cards is 50%.
The reward on 3 cards is 33%.
The reward on 4 cards is 25%. And that is the turning point. I would not take the risk here to get a 5th card at all.

If you start at 2 cards and want to go to 4 cards. Then the rewards are 50% * 33% is 16,7%. This too will be considered to low for some players. When they had success with their first probable of 50% reward bust draw. They will not risk that 33% reward bust draw.

Players will be drawing at least 1 card.
2 is the average.
4 is the psychological maximum.

I hope this information has some use.

TL;DR: You can self-balance by incorporating simultaneous turns so everyone faces the same decision at the same time; reduction of hazards the more frequently hazards affect the players; and increasing rewards for those that choose to risk by pushing their luck longer/more frequently.

More Details:
A game I absolutely love is Incan Gold, and it relies heavily on push-your-luck and self-balancing. It also uses the "draw one hazard card and it's a warning, draw a second of the same hazard card and you're out" mechanic, similar to what you describe in your original post.

It self-balances because the deck is reshuffled at the end of every round, thusly:
- the fewer players still in on a round, the more their potential reward. Those who continue to push will earn greater rewards.
- if players all stop pushing before a pair of matching hazards is drawn, then the round ends normally.
- if one player drops out before the others, the final player that successfully stops pushing collects all remaining treasures. In other words, you're rewarded more for pushing your luck more.
- if all players keep pushing until a pair of matching hazards makes them lose everything, one of that pair of hazards is removed from the deck before reshuffling and starting the next round.

There are also some other dynamics at play, including unique treasures that are won when only a single player pulls out. It's a fascinating game that does a lot with the push-your-luck mechanics, and I thoroughly enjoy playing it with large groups (up to 8 players).

See it here, should you be interested:
https://boardgamegeek.com/boardgame/37759/diamant

Never played Port Royal, if I recall correctly, so unfortunately I can't comment on that here.

If those 4 cards are a nice spot.
Then the rewards are just right.
If you want to increase the rewards. You need to lower the chances of busting. But the 4 cards will be higher as well. If the cards are the rewards, then this can be your goal. You can increase chances by reducing the number of red and blue cards. Or, you bust a player when it has 3 red or 3 blue cards.

If everyone has the exact same chances, the game is balanced. However, a deck is limited, not infinite. Especially if one player picks all its cards. And a lot of white cards are picked. Then the next player has less chance having success. This will balance itself as well, since that player knows his/her chances are lower. But only that would be the unfair part. Maybe you can have all players pick 1 card before they pick again. That way, the chances are somewhat more leveled out.