TIL In 2002 a 19-year-old garbage man won a 15 million dollar lottery, spent it all on drugs, gambling, and prostitutes, and became a garbage man again.

The idea that the probability changes the more heads you get is literally the Gambler's Fallacy. It does not change no matter how many heads you got before because each coin toss is an independent event. It has no memory of previous tosses.

The probability of getting 4 heads in a row that you calculated is correct, but it's literally a consequence of each toss being independent and each toss having a 50% probability.

The reason why casinos are profitable is because of the Law of Large Numbers. The expected payout always favors the house, and the payout approaches the expected payout the longer one plays. Basically, in your example, it's not unrealistic to get 4 heads in a row with 4 tosses (100% heads) instead of 2 heads and 2 tails. But after doing 1,000 tosses, there will be close to 500 heads and 500 tails... After 10,000, it will be even closer to 5,000 heads and 5,000 tails, approaching (50% of each).

The next toss will always have a 50% probability of getting heads or tails, even if you just got 10 heads in a row. Otherwise, the coin tosses are not independent events.

/r/todayilearned Thread Parent Link - en.wikipedia.org