Assume a blackjack player has about a 48% chance to win each hand, on average. I just had a streak of 21 non-winning hands in a row (true story!) and I’m trying to wrap my head around just how unlikely that streak was. Anyone willing to do the math, or at least tell me how to do it? Is it possible that I just had the single worst possible blackjack streak of all time? It kind of seems approximately like throwing 21 heads in a row on a fair coin.
So, very simply, the probability of losing a hand is (1-probability of winning) = (1-.48) = .52.
The probability of a streak of the same event occurring is (probability of event)[sup]Number of times[/sup] or .52[sup]21[/sup]. That comes out to around .0000010866, or a little over one in a million.
That’s a bit over double the probability of getting 21 heads in a row, which would be .5[sup]21[/sup] = .0000004768, or about half of one in a million. (That extra .02 adds up over time!) Kind of shows how quickly the extra 2% accumulates for the house.
That’s assuming that you’ve got a constant 48% chance to win. It disregards some relevant factors. For example, if I were writing a simulation, your situation would require reshuffling after every hand. Since you probably didn’t reshuffle after every hand, the actual probability is going to vary quite a bit depending on how rich or poor the deck is.
If we assume that a blackjack player wins each hand independently of the others with probability .48, then yeah, losing 21 hands in a row is only slightly more likely than flipping heads on a fair coin 21 times in a row.
However, there are a couple important caveats:
First, you’ve probably played many more than 21 hands of blackjack in your life. With the same assumptions as above, there’s some point at which not losing 21 hands in a row becomes highly unlikely. I don’t have a good estimate for what that value is, so I can’t say whether that’s a reasonable explanation.
Secondly, there are a few assumptions here: plays are independently and identically distributed, and all blackjack players have the same probability of winning any given hand. These are, to put it mildly, complete bullshit. There’s enough strategy in blackjack that skill does matter, and given what we know about people being willing to take big chances to recover their losses, it’s very likely that either your chance of losing a hand is higher than you think, or that you were a much worse player on the 21st hand than you were on the first.
And here I thought 11 losses in a row on my last trip was bad. Could always be worse, eh?
Are there blackjack simulators out there that account for different variables (e.g. number of decks, frequency of reshuffling) and different parameters (e.g. players hit as if they are a dealer, basic strategy, random)?
You know, I’m not sure. I’ve never written one myself. I’d be surprised if there wasn’t one on the web, but I don’t know where you’d find one. 
Blackjack isn’t random. It’s also includes human decision making. A person that plays stupidly could almost guarantee a loss every hand. Gee I’ve only got 20 I’ll take another card.
Don’t take this the wrong way, but the 48% is based on playing the best possible strategy. Have you considered the possibility you might be misplaying some hands?
Many players, for example, don’t realize there are situations where you should stand on a 12 because mathematically it’s better odds.
Optimal strategy can reduce the house edge to as low as .02%. These games are hard to find. Typical “strip” rules lead to an approximate .50% house edge.
FWIW, one only stands on a hard 12 with a dealer showing 4-6
The Wizard of Odds has covered a very similar question, the odds of a losing streak of 19 hands in a row.
It should be noted this assumes “starting on the next hand from now…” getting that many losses in a row. Over a long session with thousands of hands, the odds are better that at some point during 19 of those hands there will be such a streak. (I think…)
Yeah, everyone saying that the situation isn’t realistic is correct. If it wasn’t clear, my answer was for a situation specified in the OP, where by assumption the player has a .48 chance, at random, of winning each hand. That’s closely modeled by an excellent player playing perfect strategy at all times. Whether or not that applies in the OP’s case is unknowable at this point. 
This looks about correct even with optimal play. (That Win is much less likely than Lose is compensated by higher payoffs for Natural 21, Doubling-Down on 11, etc.) OP didn’t mention ties; perhaps he ignored them in which case the loss rate is 53.6% even with best play.
As others suggest, 21 heads in a row is unusual but hardly unheard-of. You need to get to streaks of 26+ before you consider contacting the Guinness Book of Records. “On August 18, 1913, at the casino in Monte Carlo, black came up a record twenty-six times in succession in roulette.” (Googling shows 32 consecutive reds in a Brazilian casino, … fake?)
And be aware (though I have no personal evidence of this that would stand up in a court of law) that the card sequences dealt by some dealers sometimes do not follow the expected random distribution. :eek:
I was trying to count only non-wins. I was including ties in the category of “non-winning” hands. I had 2 pushes during this 21 hand non-winning streak. According to crazyhorse, a “win” will only occur about 42-43% of the time, though, so it looks like a 48% expected win rate was quite optimistic.
I was using optimal play strategy. I didn’t actually make any decisions during my play but rather followed basic strategy, so the human element of bad play Little Nemo and others have wondered about can be ignored. The table rules were dealer hits soft 17, dealt from a 6 deck shoe.
I see that now, since it seems like even with conservative assumptions, any particular streak of 21 hands has a 1 in a million chance to end up like mine. There are easily a million cases of 21 hands being dealt to a person playing blackjack every week, if you consider that if a player plays 30 hands, he had 10 separate streaks of 21 hands being dealt to him. Am I looking at the situation from the right perspective now?
Even then, the ball is spun in roulette MUCH LESS frequently than a hand is dealt in blackjack. There are easily 20 blackjack players for every roulette player, and a dealer will deal blackjack easily 5 times for each time a roulette dealer spins their game.
I’m less sure about that. Are you suggesting that some dealers cheat? I’m quite sure this one wasn’t, at least.
According to the Wizard of Odds anyway - but it has always proven safe to assume his calculations highly accurate in all question of gaming odds.
Its a variable question though depending on the game rules and number of decks. His odds table for blackjack assumes a 6 deck shoe and Las Vegas strip rules (dealer hits on soft 17, etc.) and as noted the calculation is for determining the odds of a streak of X losses during exactly X number of hands. It’s much less likely to lose 21 hands in a row out of a total of 21 hands dealt than it would be to lose 21 hands in a row anywhere within a larger session of, say 100,000 hands.
Best possible strategy will get you closer to 49.5% around here. In Ontario the house stands on soft 17, which helps the player, and there are some other favourable rules.
However, losing 12 in a row is pretty damned unlikely. Even if someone is not playing absolutely perfect strategy their decisions that aren’t right will usually be on the margins, not stupid shit like standing on 6.
I have lost nine hands in a row. Just wiped off the table; as I recall I barely had a decision to make.
Did you ask this question to the dealer/s? No, yes, I’m serious.
Maybe the place you were playing was crooked. If so, shame on everyone. However, if that wasn’t the case, the dealer/s have probably seen so much action that they would, or might, willingly give you a tip or two on your strategy. They don’t make more money with you losing but if you win they may get a nice tip.
Can you think of a few hands in that run where a more conservative play would have given you a win?
If the place is on the level, the dealers are impartial or even if favor of the players. The casino will teach you how to play and/or help you be a better player. The odds are already in their favor, they don’t want to drive customers away. The whole thing is to get the customers in and keep them playing as long as possible.
49.5% might be the expected return, but as I have learned in this thread, expected return is NOT win percentage, because many actions can increase your expected return while not improving your win percentage. For example, doubling down doesn’t increase your chance to win that hand (and in fact might lower it, since there will be a few times during an appropriate double down where you’d wish you could have hit twice) but greatly increases your return, by giving you extra money in favorable conditions.
You’re probably correct, but why are you quite sure? When I played a significant amount of Blackjack in Las Vegas 30 years ago, cheating was much more common than most outsiders would have thought; this was confirmed by friends of mine in the gaming industry.
I could relate interesting anecdotes but they’d be denied here … by the very same people still claiming player’s winning-hand percentage is 48% or 49% despite three Dopers now pointing out why this is wrong!
I find it unlikely dealers will cheat, especially in favor of the house. Dealers make their money off tips, and people who are losing don’t tip dealers.
I’d be very interested to hear those anecdotes!