The probability of the outcome is exactly the same every time you flip it, so the probability of being right is exactly the same. The reason it looks different is that there are many flips. If you flip it a billion times, it will look like the 9.5% is wrong by the time you’ve done that. But in fact, it’s right, only the probability of being right depends on how many flips have gone before.
The same way is true with the coin toss in Nock’s analogy. If I get one in a row, it will seem as if I have a much higher chance of being right on this one, and the probability of being right is higher. If I get two in a row, I’ll still see the odds as higher, because I saw one in a row in the previous one. In fact, the odds are the same, because I’m just flipping it a billion times. The reason that it seems like I have a much higher chance is because of all the previous coin tosses. It’s easy to imagine how you can get a bad habit of thinking in this way. If you see two flips in a row that come up heads, you’ll start feeling like it’s an even money coin, and that’s what the odds are. But you’ve only been flipping it a billion times. The fact that you’ve seen one in a row doesn’t mean that you’ll continue to see one in a row. If you’ve flipped it a billion times before, you’ve probably gotten a bad habit of thinking about how you’ll do.
To get a good habit of thinking in a more realistic way, you have to take the coin out of the box and toss it a bunch of times. (You can even spend more time with it by doing mental tricks to get yourself to do it, as I’ll describe later.) The results of each tosses will shift your perspective, and will make it easier for you to see that each toss is the same coin as the previous one. As you do this, you’ll also realize that the “result” of each toss is actually random. Even if you have one in a row, it’s not a sure thing, even if the outcome is a heads.
And if you go to the Olympics and see a coin flip that comes up heads, you might assume that this means that the next coin is guaranteed to flip heads, but that’s not true at all. In fact, flipping a coin a bunch of times, one way or another, always has an equal probability of coming up heads.
Similarly, when we do experiments in a lab, we give up a little bit of certainty on the outcome of each experiment. We want to be sure that each of the results we get are 100% accurate. But when you think about it, you shouldn’t have to worry about the results of experiments, because we only consider each outcome separately, and each outcome has an equal probability of being the real answer. You should also not consider the results of a single experiment to be very important. When we do an experiment, we first evaluate the question, and then we evaluate how likely each answer is, and then we see how likely we are to get the answer that we want. The probabilities we give the answers are meaningless in this context.
The reason that the answers we give are not meaningful is that each of the answers is a single coin flip.