One factor that makes probability hard to understand on an intuitive level is that our human brains work against us to some extent. We have a natural tendency to look for patterns and to assume that patterns are meaningful even when they are random. Or aternatively to overlook meaningful patterns because they don't intuitively look like patterns.

In real life, that is our own personal lives, seemingly random events have occurred that have
resulted in profound changes to us. We see the events as random because we lack knowledge,
even though they aren't random.

In real life, that is our own personal lives, seemingly random events have occurred that have
resulted in profound changes to us. We see the events as random because we lack knowledge,
even though they aren't random.

One factor that makes probability hard to understand on an intuitive level is that our human brains work against us to some extent. We have a natural tendency to look for patterns and to assume that patterns are meaningful even when they are random. Or aternatively to overlook meaningful patterns because they don't intuitively look like patterns.

Going back to the coin toss example, for instance, we might see a sequence of heads as a meaningful pattern. Sequences of heads thus "pop out" at us when reviewing a series of coin toss results. Any large sequence of heads intuitively seems unlikely and remarkable.

Meanwhile, we overlook patterns that are equally unlikely as any given sequence of heads. A sequence like H-H-T-H-H-T-T-T-H-T is equally likely as a sequence of ten heads in a row. But there's nothing about that first sequence that is easy for us to grasp, so we basically just ignore it, maybe file it away in the category of non-meaningful noise.

This is a failure of human perception. Our minds are constantly doing this to us, reinforcing some patterns, overlooking others.

...Meanwhile, we overlook patterns that are equally unlikely as any given sequence of heads. A sequence like H-H-T-H-H-T-T-T-H-T is equally likely as a sequence of ten heads in a row...

Actually, a sequence of 10 heads in a row is much less likely than the sequence you posted. It's very easy to calculate. On a simple coin toss example, the odds of 10 heads consecutively would be less than 1 percent (.00097), which would mean that over the course of several 10 series samples, you'd expect to see 10 consecutive same outcome strings about 1 out of 100 times (or once out of 1000 flips).

Actually, a sequence of 10 heads in a row is much less likely than the sequence you posted. It's very easy to calculate. On a simple coin toss example, the odds of 10 heads consecutively would be less than 1 percent (.00097), which would mean that over the course of several 10 series samples, you'd expect to see 10 consecutive same outcome strings about 1 out of 100 times (or once out of 1000 flips).

The first quoted post is correct. There is nothing special about heads. Assuming your calculation of .00097 to be correct (I haven't verified it), the odds of HHHHHHHHHH and the odds of HHTHHTTTHT (or any other specific sequence) are both .00097.

But how many times would you expect to see this exact sequence:

H-H-T-H-H-T-T-T-H-T

Consider that the sequence must occur precisely in the order posted.

I think you'll find that it's actually the exact same probability as 10 consecutive heads. In both cases, you need each flip to match the predetermined pattern precisely.

To think of it another way, the odds of successfully "calling" 10 tosses in a row are the same if you call heads each time, or if you vary your call between heads and tails. In either case, you're equally unlikely to get each call correct 10 times in a row.

I'd posit that the reason we tend to see the H-H-T-H-H-T-T-T-H-T sequence as more likely (even though it's not) is that we assign no particular "meaning" to H-H-T-H-H-T-T-T-H-T, which makes it harder for our brains to spot it.

I agree that determining whether something is random requires a different perspective.

In my view, people are hard-wired to view the world as meaningful or purposeful, which makes it hard to judge what's random and what isn't. If event X happens, we might view that as a "sign"; we might believe that event X was meant to be, or be glad that event X happened because it took our lives in one particular direction rather than another.

The big question to me is whether we would feel exactly the same way if event Y had occurred instead of event X. The exact opposite of event X even. If event Y happens, our lives are diverted onto a completely different path, but I'd wager that this path will seem no less meaningful (i.e., non-random) than the path that would have been triggered by event X.

If this is true, then it means that our own perception of the meaningfulness of a particular event can't be trusted. The event might be random, or it might not be; we need a different metric outside our own perception to figure that out.

Humans are meaning-creating machines.