I am not a math guy, so here goes…
And no this has nothing to do with a Martingale system.
Say you have a 1 in 4 chance of winning a bet. If you lose 3 times in a row, what are the odds you will lose four in a row? Is it still 1 in 4? Intuitively I feel no it isn’t, but each time you have exactly a 1 in 4 chance of losing.
Hope this makes sense enough to get an answer. 
It depends on what you’re betting on. If you’re throwing say a four sided die, then each throw is independant and each time you have a 1 in 4 chance of winning. However, if I lay out four cards, say an A, 2, 3 and a 4, and you decide to go with the Ace, you draw a card with a 1 in 4 shot. Then I let you pick again, and you have a 1 in 3 shot, then I let you pick again, and you have a 1 in 2 shot, and so on.
But generally if you’re betting on something, it’s going to be independant of the last bet. Your way of thinking is commonly called the gambler’s fallacy.
To clarify: in the above situation with the cards, each time you picked a card, it was removed from the selection, leaving only 3, 2, and 1 remaining cards respectively.
That sentence is correct. The odds of a radnom outcome do not depend on prior outcomes.
In other words, dice have no memory.
Are the odds different for losing 4 times in a row then? ie you count previous occurences.
Don’t feel alone mchapman. Most of us struggle with this concept. The classic example is flipping a coin. Assuming an unbiased flipper you have 50/50 chance of either heads or tails coming up on any given flip. Let’s say you flip 99 times and for some strange reason every time it comes up heads. It seems almost inconceivable you still only have a 50/50 chance on the 100’th flip coming up tails… but those are the true odds. The idea is that if you were to flip the coin enough times that eventually it would even itself out to around 50/50.
I know that it’s not intuitive, but you have to accept that odds don’t change depending on what has happened in the past…
Are you asking what the odds are for losing 4 times in a row, or for losing the 4th time? For the first one, assuming you’re rolling a 4 sided die or something, you’d have to multiply the probabilities:
3/43/43/4*3/4 = 81/256, or about 31.6%.
Note that the probability of winning 4 times in a row isn’t 1-81/256, because in this case, there are more outcomes than just winning 4 times in a row or losing 4 times in a row.
Winning 4 times in a row:
1/41/41/4*1/4 = 1/256, or about 0.4%
If you’re wondering what the chance of losing the 4th time is, it’s still 3/4, because again, dice have no memory.
Or, are you asking “what are the odds of losing 4 times in a row GIVEN that you’ve already lost 3 in a row?”
Is that case, it’s just the chance of losing once. . .3/4.
If you want to talk about the a priori priority of losing 4 times in a row starting from 0, that’s what audiobottle was saying.
Thanks audiobottle got it. Sorry to be confusing.
If the events you’re gambling on are independent, then everyone else’s answer is correct: the probability that you lose the fourth bet doesn’t depend on the outcomes of previous bets. As far as I know, all the common casino games have this property.
If the events you’re gambling on are not independent, then there’s not enough information to answer the question. A given athletic team’s probability of winning their next game probably does depend on the outcome of the past few games, so don’t count on independence when you’re placing sports bets.
There are some other considerations based on how likely the events you’re betting against are. If, for instance, you’re betting that the faces of two dice won’t sum to 12, and you’ve lost three bets in a row, you should reconsider your hypothesis that the dice aren’t loaded. But that’s complicated, and highly dependent on the exact nature of the bet.
Your intuitive feeling is, as Electronic Chaos pointed out, common enough to have a name: the Gambler’s Fallacy: the feeling that, after a long string of losses, you’re “due” for a win (or vice versa) when each result is really independent of all the ones before it. I’ve read speculations that we tend to think this way because, in the natural world in which human intuition evolved, very few events truly are completely independent.
This smells like the Lets Make a Deal thread. Are we going to go there?
I am also uneasy about this question
I also occasionally like roulette - but only when I set a limit.
To me it seems that the odds of the fifth roll coming up red after four reds is pretty strange, the events might be unrelated, but the sequence is related.
Well, one thing unmentioned here is the assumption that you are using a fair die. From a purely theoretical standpoint of course you can theorizes a fair die, 1/6 probability of any side coming up. But in the real world we are rarely given a fair anything, even a coin might have the slightest tendency toward heads over tails, for example. Back to the die we have a a priori expectation that the die is fair. So a one comes up and we think noting of it. Then a one comes up again and we thing, interesting, but just random chance. Now it comes up a third time and the teeniest suspicion creeps in. Now a fourth roll produces yet another one and your suspicion grows. So now you are not unjustified in possibly thinking the fifth roll has a higher prbability of producing a one. The theoretical die, no it has no change in probability. But in real trials, yes possibly so. This process is known as Bayesian Inference or Bayesian updating.
So you are not way off base with the concept that it seems strange that another one shouldn’t be more likely. If you rolled the die 99 times and got one every time? Well I would seriously consder an even money bet it would be one again.
My main question was what are the odds of winning/losing four times in a row when each individual betting instance had a 1 in 4 chance of losing.
Hope this explains my question better. Audiobottle answered my main question.
The intuition thing is hard to get by, I wonder if there is any good research into why people feel this way?
rchapman (and others), consider the simpler case of coin flips with a fair coin. It seems exceedingly improbable that I can flip, say , 100 Heads in a row. Surely, after flipping 99 Heads, the odds of flipping that 100th must be smaller. But the odds on that particular flip are, indeed, 50% either way.
But what about the odds of that long a run? The probability of getting 99 Heads in a row is (1/2)^99. Multiply that by the (1/2) probability of getting a 100th Head, and you get (1/2)^100. But what’s the peropbability of getting 99 Heads followed by Tails? It’s (1/2)^100 as well! The “hard work” has already been done by the previous 99 flips, you might say. The truth is that it’s just as improbabl;e to get any particular sequence of results – HTHHTHHHTTTTTHTHHTTTHHT…H as any other. The result of all heads just stands out as particularly noticeable, and one with a grotesque preponderance of one side over the other.