Counting the ways an outcome can happen
Three fair coin tosses make eight patterns. Exactly two heads is three of them, or 3/8, not a specific order and not a system.
Counting the ways is how a small chance becomes a fraction you can check. If each trial has two results and the trials do not affect each other, the number of patterns with a given count is a combination. Multiply that count by the chance of one pattern and you have the chance of the count. The method is old. It still gets misused as a reason to stake.
This is for adults 18 and older. The coins below are fair. The article is not a system. Laws differ by place.
Three tosses, listed
A fair coin tossed three times has eight equally likely patterns: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Each has probability 1/8, because 1/2 times 1/2 times 1/2 is 1/8.
Exactly three heads: one pattern, probability 1/8. Exactly zero heads: one pattern, probability 1/8. Exactly one head: HTT, THT, TTH, three patterns, probability 3/8. Exactly two heads: HHT, HTH, THH, three patterns, probability 3/8. The four probabilities sum to 1. If your counts do not sum to 1, you missed a pattern or double-counted one.
The three patterns for exactly two heads are the combination C(3, 2), which is 3. Combinations ignore order inside the hand of results. HHT and HTH are different sequences and the same count of heads. The combination counts the sequences for you so you do not have to list them once the number of trials grows.
Why the list beats a hunch
People say "two heads out of three is pretty likely" and then treat a specific order, heads then heads then tails, as if it had that same chance. The specific order is 1/8. The count "exactly two" is 3/8. The tree keeps the order. The combination collapses order when you truly do not care about it. Know which question you asked.
The same collapse is where parlays go wrong in the other direction. A parlay needs every leg in one specific pattern of wins. It does not get to claim the more generous "somewhere in the card" count. A parlay is less likely than it feels because it is one path, multiplied.
What the count does not say
It does not say a coin that has produced one head in three tosses is due two more. The eight patterns describe the tosses before they happen. After you have seen them, the next toss is a new trial. Streaks and the gambler's fallacy live in that confusion between a finished count and the next flip.
It also does not say the game paying you is fair. Three heads at 1/8 is the chance. If someone offers less than 7 to 1 on that specific pattern, the offer is worse than the count. A correct count of ways is not a wage. House edge is what you compare with the count, not a substitute for it.
Keep the trials matched
The eight-pattern list requires identical, independent trials. Cards dealt without replacement are not that list. Sports games are not that list. If the chance changes from step to step, go back to a tree and write each branch. Forcing a coin formula onto a shoe or a season produces a precise wrong answer, which is worse than a vague right one.
Use the count to check a claim. Then stop. If an elegant 3/8 is becoming a reason to buy a ticket, the elegance has done enough. A limit you set in advance still governs. If counting is how sessions grow, close the page and use local help if money is already involved.
Frequently asked questions
Is exactly two heads in three tosses a 50 percent chance? No. There are three patterns out of eight, so the chance is 3/8, or 37.5 percent. A specific order, such as heads-heads-tails, is only 1/8.
Do the pattern chances sum to 1? Yes. Zero, one, two and three heads are 1/8, 3/8, 3/8 and 1/8. They cover every pattern once.
Can I use this count for a slot or a sports parlay? Only if the trials really are identical and independent, and only if you want that exact count of hits. Most parlays and most games fail one of those conditions. A wrong model with a tidy fraction is still wrong.