Hmm…I understand the math of this, but I’ve always had trouble rationalizing it intuitively (as with a lot of probability theory).
I understand that the coin has no memory, so that for each flip there is a .5 probability of getting H or T (assuming a perfectly fair coin), regardless of previous outcomes.
Also, I understand that even if you’ve flipped this perfectly fair coin and got 9 Hs, the probability on your 10th flip of getting an H is still 0.5.
I understand mathematically that if you decide ahead of time that you want 10 Hs in a row, the probability of that is (.5)[sup]10[/sup] - i.e., something small, much < .5.
I just can’t rationalize this intuitively - I’m grasping for a visceral feel for it but coming up short. My problem with it is precisely because each flip of the coin is independent of the other flips, and the coin has no “memory”. Why should 10 Hs in a row be any less probably than any other sequence? The coin has no preference. I had the same trouble in high school math, with problems like the probability of rolling a 3 and a 3 with two dice, compared to, say, a 3 and a 4. I know there’s only one way of getting 3-3, and 2 ways of getting 3-4, but intuitively the dice shouldn’t be able to discriminate because the 3s happen to look the same. I guess there are two things to be gleaned from this: (a) intuition is not always a good guide in math, especially probability and (b) I’m not able to articulate very well what I don’t like about these answers.
Yes, I have actually tried experiments with coins and dice, both real and simulated, to convince myself of these numbers. Still doesn’t quite feel right, though. There seems to be something essentially “ungraspable” about probability, whereby you just have to trust the formulas.