A little thought will convince you that in this case, with a starting count of zero, we always have an expectation of 0 when the running count is +4 (indicating that the number of white and black balls remaining is equal). Hence, the pivot point for this game is +4. Further reflection will reveal that we now need the running count to be equal to or greater than +5 to have the advantage. The key count for this game is therefore +5. As you can see, both the key count and the pivot point changed in response to altered starting conditions.

Since this can all become a bit unwieldy, we can, if we choose, make an adjustment in the point at which we begin our count. That is, we adjust the "initial running count" (IRC) in order to provide more convenient key-count and pivot-point numbers. For example, we could start the IRC at, say, +2, and then we'd have the advantage when the running count was equal to or greater than +7. Or we could use an IRC of -4, in which case we'd have the advantage when the count was equal to or greater than +1.

The point of all this is that the pivot point and key count are a function of what we choose as the IRC. We'll see later how this can be used to simplify our system.
Let's review.

• To get an advantage in our gumball game, we can assign an integer value to each colored gumball and keep a running count of those we've seen come out of the machine.
• Two special count values are the key count, at or above which we have the advantage, and the pivot point, at which we have reliable information about our expectation.
• The key count and pivot point will depend on our initial running count.

Counting cards is not dissimilar to counting gumballs. First, just as there were good gumballs, bad gumballs, and neutral gumballs, in blackjack there are good cards, bad cards, and neutral cards. And just as we assigned a value to the different gumballs, we can also assign a value to each type of card. We then start with an initial running count and count through the deck as the cards are played. Once the running count reaches the key count. we know we generally have the advantage. When the running count is equal to the pivot point, we have a reliable estimate of the expectation.