I recently had a disagreement with a friend about what it means for an event to have a probability of happening every X times.
Take for example a coin that if flipped has a 50% probability of landing heads and a 50% probability of landing tails.
My understanding is that while the probability of landing tails 3 consecutive times is smaller than that of landing it 2 consecutive times, if in the last two flips you got tails it has no effect on the probability of you getting tails in the following coin flip.it stays 50-50.
But he seems to believe that the probability of you getting tails gets smaller and smaller the more you land multiple consecutive tails.which sounds very counter intuitive to me.
For a more concrete example, he seems to think that if an event has a probability of occuring every 10 years, and it’s been more than 10 years since the last time it occured, we are living on borrowed time and the probability of it happening now is extremely high while I think it had the same probability of occuring today than it had of occuring 9 years ago.
Which one of us is wrong, please give examples.
In: 5
The chance is the same, but the probability of it happening is higher. Still 50/50 for the coin to land on heads or tails, but the probability of one or the other is higher. In the 10 year case, the chance of it happening is the same today as 9 years ago, but the probability of it happening is higher based on how long it has been.
You are right.
The probability of flipping 3 tails given that you’ve flipped 2 tails already is 50%.
Think about it physically for a moment – how can a coin possibly remember what the last flips were? What physical process would cause it to be less likely to land tails based on past behaviour?
If you flip a coin ONCE, you have a 1:2 probability of landing “heads”.
If you flip a coin a second time, you still have a 1:2 probability of landing “heads”.
HOWEVER.
Your probability of landing “heads” with two successive flip is 1:2 multiplied by 1:2, for an overall probability of 1:4.
Thus, your probability of landing three “heads” in succession is 1:8 (1:2 x 1:2 x 1:2).
For the coin toss, you’re right. Each coin toss is independent from the previous one. You have a 50% chance of getting tails each time. Imagine you tossed the coin 3 times but hid the results from him. Then he tossed it once. Does he have a lower chance of getting heads just because your previous tosses were all tails? Does him knowing whether they were heads or tails change how the coin flips in the air? Does the coin have a memory of all the times it’s been tossed in the past? If not, then by what mechanism is it going to know that it did heads the past 3 tosses and should do tails now? It doesn’t really make any sense to think that the previous events here could affect future events. Each time you throw the coin up there’s a 50/50 chance it will land on either side.
With the event that happens every 10 years it’s a bit trickier because that’s an expected value not a probability. Without more context, we don’t know how that expected value was calculated, so it could be imminent or it could be random depending on what it is. For example, imagine you’re filling a cup with water and it generally takes 5 seconds to fill it up. If it’s currently been 6 seconds, the chance that the cup will overflow is imminent, and is increasing over time, and so in that case you are on “borrowed time”. But if it’s something like a weather event or something random that isn’t directly tied to duration since the last event then you’re not on borrowed time for that. The 10 year value is just based on averages, not how long it takes to build up to the event. So it depends entirely on what it is and why the event is likely to occur “every 10 years”.
Your friend is falling into the classic *gambler’s fallacy.* The coin has no memory of past flips, just like the universe doesn’t keep track of how long it’s been since an unlikely random event.
Independent events are independent.
Coin flips are generally seen as independent events. You can find the probability of multiple independent events happening in series, that is your 3 tails in a row (12.5% chance), but each independent event is independent.
That is to say, if you are given 2 tails in a row, your chance of tails on the third flip is still 50%. The difference between the 12.5% and the 50% comes from the fact that saying “I just got two tails in a row” only happens 25% of the time. You will note that 50% of 25% is in fact 12.5%.
Each individual coin flip is utterly unaffected by the preceding flips, and has no effect on future flips. Thus, any and every given coin flip has a 50-50 outcome, assuming a fair coin.
The probably of an event depends on what the event is. If you are asking what is the probably that you get 3 tails in a row its 0.5³ so 0.125, 12.5%. But the coin flips themselves are independent so if you had two tails the probably that the third one will be tails too is 50%. Getting those two tails in a row already had a probably of 25% but that happened. Probably is about asking what can happen in the future and independent coin flips dont effect each other.
Its always good to write out the possibilities:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
If you got TT for the first two you will either get the TTH or the TTT scenario. 50% probability for each.
From the start the TTT scenario is 1/8 of all the possibilities, getting either of those scenarios are equally likely but when you already got two tails 6/8 of those scenarios are no longer possible so that leavs 50-50 odds.
You are generally right. It depends on the circumstances of the circumstances of the problem.
If one event has an impact on the next even the next event, then his reasoning is good. An example of this would be drawing red balls out of a bag instead of yellow. There is a finite number of red balls in the bag, ao every time you draw another one out, the odds get higher against it happening again.
However if prior events have no impact on the next event, like your example of a coin flip, then the odds don’t change, so it would stay 50/50 as in your example.
If he still doesn’t understand it, ask him to explain to you how the previous coin flip physically impacts this coin flip. He won’t be able to do it.
Coin flips are independent events. The values of previous coin flips do not affect the values of future coin flips. Believing otherwise is known as the Gambler’s Fallacy.
Some events are not independent or are otherwise cyclical. In that case, knowing they have not happened for a long time can tell us something about the probability that they will happen now. Say I select my socks every morning at random and have one pair of purple socks. Each morning I select a non-purple pair of socks, the probability of selecting purple socks the next morning goes up (conversely, if I pick purple socks this morning, my chance of picking them next morning craters to 0). This doesn’t violate the Gambler’s Fallacy, because my sock selections are not independent – if I wear a pair one day, I can’t wear them some other day.
So the trick is to think about whether the events are independent or not. Most of the things we do in games of chance (flip coins, draw cards, generate random numbers, etc.) are independent. Many of the things we deal with in real life are not. The Gambler’s Fallacy mostly arises from us taking our real-world instincts into a special place (like a casino) where they don’t apply.
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