Coin flipping - likely to hit $7$ heads or tails in a row (answered) and betting progression

I am editing this question as requested so I am more clear in what I am asking.

Assuming that a coin flipped has a $50\%$ chance of landing heads and a $50\%$ chance of landing tails, I had wondered how many times I would have to flip the coin on average to end up with specifically $7$ tails in a row. The answer I think is that after $128$ flips I would run the risk of getting $7$ tails in a row, and after $254$ flips I could be expecting to get $7$ tails in a row.

The second part of my question was, if I bet on my coin flips trying to get heads and I started with a $5$ dollar bet and used a betting progression that looked like $5\$\rightarrow5\$\rightarrow5\$\rightarrow30\$\rightarrow100\$\rightarrow300\$\rightarrow500\$$ for the first through seventh coin flips in such a way that any time I flipped heads and won I would start the progression over, and any time I flipped tails I would advance to the next step in the progression order and try to get heads, then how many flips can I make before I would be more likely to start losing more than I gain, or after how many flips would it be advisable to keep what I have won so far? (edit: if there was a $60\%$ chance to land tails each flip)

Everything below this point is my original question, above is my reworded question. I took out all the blackjack related aspects as its way too complicated to figure odds without building a specific program that follows my personal blackjack strategy on top of betting progression.

First off I have a more simple problem, and then a more involved complex problem.

The simple problem is, if I flip a coin over and over, how many times do I need to flip it before I'm likely to end up with $7$ tails in a row?

From what I calculated by adding up the number of times I could flip heads or tails, is that in every $7$ flips there are $254$ heads and tails possible. And if only one of those combinations is tails $7$ times in a row I took $1$ and divided it by $254$ and ended up with $0.0039370078740157$ (edit=actual number is $.0078126$) chance of getting $7$ tails in a row on any given $7$ flips. Or should it be $1$ divided by $128$ because by the seventh flip there are $128$ possibilities and only one of them as a seventh tail?

And I think that if I flipped coins over and over I would have to flip it $128$(edited) times before I would be likely to get $7$ tails in a row but I'm not sure.

Now the second more involved question is:

If I play blackjack and there are 8 decks and I double down on $11$, split on $2$(being a pair of aces), $6, 7, 8,$ or $9$ and otherwise always stand on $12$ or higher, while betting a progression bet that looks like $5\$\rightarrow5\$\rightarrow5\$\rightarrow30\$\rightarrow100\$\rightarrow300\$\rightarrow500\$$ returning to the initial $5\$$ dollar bet after any win, with an initial bankroll of $1000\$$, what are my chances of losing all my money If I play until I have $7000\$$? While the dealer must hit on $16$ or soft $17$, push if we both get blackjack, but otherwise the dealer wins if he has blackjack before I can hit. (house rules I found played near me that are helpful to the player: can hit to reach $21$ and still get blackjack payout bonus, can hit after splitting aces, can double down after hitting) Unsure if splitting ten value cards is productive, could get $21$ or another $20$ and seems risky

I would really appreciate help clearing up the myth that betting progression systems do not work, as most people refer to the martingale system where you just double your last bet until you win, my system is more complex because you watch $3$ small $5\$$ dollar hands for a string of $3$ losses then start ramping up the bet a lot until you win.

3 Answers

Regarding the coins, the expected time to get $7$ heads in a row is $254.$ The time to get a specific generic length n sequence of heads and tails (like HTHHTHHHTTHTT) is about $2^n$ (=128 in your case) but the time to get strings that have a subsequence at the beginning that is the same as the one at the end is longer. The string of all heads (or all tails) is the worst and takes $2^{n+1}-2.$

Regarding the blackjack strategy, I don't know the edge for blackjack so I can't calculate that. (And I wouldn't want to even if I did.

What I can tell you is:

1. If you play any progression strategy like that with negative edge and stop when you've lost a fixed amount of money, you will lose money on average. This is a theorem. Progression systems can only work (in the sense that they make money on average) if you can weather infinite downswings.

2. If you have negative edge and you play smaller bets until you lose or win a fixed amount of money, then you have a worse chance of making money the smaller your bets are. This is not to say you ever make money on average. But you will make money in more individual sessions and you stand a chance of getting lucky. Whereas with small bets the law of large numbers takes over and you lose money all the time.

The second part of my question about how many coin flips I can make, (or hands of blackjack played) before I am likely to lose 7 times in a row betting on the 40% odd heads, (or my blackjack hand) is in my estimate 50 flips or hands. And since I only really lose money if I lose 7 bets in a row, and blackjack shoes run out before 50 hands are played on full tables, I could sit at one blackjack table for many hours slowly accumulating chips.

I personally prefer what is referred to as the 3rd base position, at the right hand of the dealer. Because all of the random actions the 2 or more players to my right ensure breaking up long strings of cards that exist in the shoe after the cards are dealt. And my decision about hitting, standing, splitting, or doubling down is all derived from the 2 cards I am dealt. So whatever 3rd card I get, or leave for the dealer to hit with is random.

Previously I had left the casino only after I lost my initial 1000, or gained 6000 or so. But I think it would be better to try for simply 3000 profit or 1000 loss each trip to the casino. Maybe I will be lucky again in the future.

I read some interesting things about a lot of math topics based off your responses and found the gamblers ruin Wikipedia page and related pages informative.

For anyone interested in gambling, or who do gamble at times, its worth checking out.

the chance of getting a head would be 1/2 so 1/2^6 = 1/64, so the chance would be 1/64. the reason I did 1/2^6 instead of 1/2^7 was because it doesn't matter if the first ones a head or a tail.