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 coin flip and the “every 10 years” might be different things and the same rule might not apply to both of them.
For the coin flip you’re right because all that matters is the *unknown* coin flips that haven’t happened yet. Once some of the flips are in the past, they no longer affect the probability of the next flip. If you *don’t know yet* what the 3 flips are, then 3 flips are only 1/8 likely to all be heads. But if you *already know* that 2 of them are heads, then the remaining one flip is a normal flip like any other and it’s 50/50. Essentially, the fact the 2 of the flips are now in the past and have already been heads, that has set their probability of being heads to a constant of 1. Instead of (1/2) * (1/2) * (1/2) you now have 1 * 1 * (1/2).
But the event with a probability of “once every 10 years” might not be like the coin flip. It could be something that gets more likely the more “aged” it becomes, and 10 years is just the mean time between events. For example, failure of parts due to fatigue is like that. “Mean time between failure” is often caused by the wear and tear of age, so something being 8 years old is different from it being 1 year old. Another example is a geyser that goes off, say, once every 2 hours on average, but with some variation. It goes off when pressure builds up to the breaking point, so the longer it’s been building up the closer to the breaking point it has become.
The coin toss is really a special case scenario because the age of the coin is irrelevant, the time spent so far is irrelevant, and so there’s no means for the past events to alter the future events.
It’s important to note that this is assuming you *already know* the coin is a ‘fair’ coin, rather than a ‘loaded’ one. If you don’t yet know that and you are *testing* to find that out, then things are different. If there’s a chance the coin could be unbalanced and biased, then a past pattern of many many heads in a row starts to increase the odds that that is the case. (Although, that wouldn’t mean it’s ‘due’ for tails, just the opposite. It would mean you’ve established that it’s likely to have a bias making it less likely to be tails.)
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