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. The individual outcomes of flips are relatively tiny, so the individual probabilities must increase as the number of individual flips increases.

On the other hand, the probability of flipping more than one outcome, being wrong about at least one outcome and being exactly right are all exactly the same: one and the same probability.

In other words, the way to understand the results is that the way the outcomes of the flips are distributed matters. But because each individual flip has a tiny probability of occurring, the probability of any outcome must be tiny, as well. So the same probability applies to both one way and the other way.

So why are these different methods for calculating the probability so radically different? I believe there is a more fundamental difference between what the probabilities represent and how they are calculated. The calculations represent the size of the range of possibilities, whereas the calculations represent the actual range of possibilities.

An Example

Imagine you are walking along a road in a village. As you walk, you pass by a man who is waving a piece of string. He is in the process of untying it, and you may or may not want to approach him. How many different string-tied packages will he have in the string?

If you see only one, you might only want to deliver it to him. If you see two, you might want to deliver it to both him and a neighboring village. If you see three, you might want to deliver it to himself and all three villages. You can figure all this out by calculating the three-dimensional volume of string on which all of the packages are attached, and dividing that volume by the number of packages in the string.

Similarly, suppose you are building a deck on a ship. You need to figure out how many rows of planks you need to build before the deck is complete. You don’t know how large the ship is, but you can estimate how long each plank is, and you know there are 1,000 long planks.

Then you ask a shipyard to place you a number of barrels containing wooden planks. Each barrel contains six planks. The number of planks in each barrel is 6,000. Now you are constructing the deck.

Each plank has to attach to one of the rows you laid before. Each row has to attach to one of the 100 feet you laid before.

If you just laid the decks one by one, you would be building decks one at a time. But because you laid them in sequence, you can make the deck any size you want. You want to make a 40 foot deck? Sure, go ahead. You want to make a 50 foot deck? Go ahead. You want to make a 60 foot deck? Go ahead. Whatever you want.

So the number of planks you need to lay in a row depends on the total number of rows you laid before it. If you had been layering the decks before, you would have to lay the same number of planks in each of the two rows to make it as big as you want.