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We will now apply our multinomial formula for counting the number of partitions to solve the following probability problem. We have a standard 52-card deck, which we deal to four persons. Each person gets 13 cards as, for example, in bridge. What is the probability that each person gets exactly one ace? Well, before we start, as always we will need a probability model. We deal the cards fairly, and this is going to be our model. But we still need to interpret our statement. To give this interpretation, let us first think of the outcomes of the experiment. What are the possible outcomes? An outcome of this experiment is a partition of the 52 cards into the four persons so that each person gets exactly 13 cards. Our statement about dealing the cards fairly will be an assumption that all partitions are equally likely. So since all partitions, all outcomes of the experiment, are equally likely, this means that we can solve a probability question by counting. We need to count ...

Now we're going to apply the same exercise that we just did. So here what we did, we showed that there is no fundamental difference between [INAUDIBLE] and English, which is [INAUDIBLE] in the history of Haitian and Creole. Because what we're doing here, we are questioning this notion that Creole formation is something which is very idiosyncratic, which separates Creoles from non-Creoles, in a fundamental sense. Now what we're going to do, we're going to look at the history of French itself from Latin. And we're going to ask, so does that history, can it be in any way compared to what happened in the history of Creole. In fact, what we are going to see is that if we compare Latin and French, in many ways, they show that French also out Creolization Creole. We're just going to do it very-- In fact, what I'm going to do now is very tendentious. But I'm warning you that it's tendentious. When I'm comparing French and Latin is exactly w...

Okay, now let's apply all this analysis to improving the performance of our circuits. The latency of a combinational logic circuit is simply its propagation delay t_PD. And the throughput is just 1/t_PD since we start processing the next input only after finishing the computation on the current input. Consider a combinational system with three components: F, G, and H, where F and G work in parallel to produce the inputs to H. Using this timing diagram we can follow the processing of a particular input value X. Sometime after X is valid and stable, the F and G modules produce their outputs F(X) and G(X). Now that the inputs to H are valid and stable, the H module will produce the system output P(X) after a delay set by the propagation delay of H. The total elapsed time from valid input to valid output is determined by the propagation delays of the component modules. Assuming we use those modules as-is, we can't make any improvements on this latency. But what about ...