# Multiplication as "of" One thing that I have run into in the past is the question: "Why do we multiply two quantities together?". For example, consider [Expected Value](Expected%20Value.md): $E[X] = \sum_x x P_X(x)$ Here, we are multiplying a specific value, $x$, times it's probability of occurrence, $P_X(x)$. But, the question remains, *why* do we multiply these two quantities? A nice way to think about this is that multiplication is the mathematical equivalent of using the word *"of"*. For example, say we are dealing with apples. In a single bushel we have 20 apples. We then have 7 bushels. How many apples do we have? Clearly the answer is $20 \times 7 = 140$. But, if we were to write this out in English it may look like: > We have **7 bushels** *of* **20 apples**. We can imagine replacing the "of" above with a $\times$, yielding our 140. So in this case we have shown that our intuitive notion of "of" is indeed mathematically equivalent to multiplication. What about fractional quantities? What about if we are dealing with pizza, and we have 8 slices to start. Then 5 slices are eaten. If we want to express the fractional amount of the pizza left we would calculate $\frac{3}{8} \times 1$. Again, we can think of this as a way of using the word "of": > We have $\frac{3}{8}$ *of* **1 pizza**. ### Expected Value This becomes important when dealing with expected value. A question I have asked is why do we simply multiply the probability of a given outcome $\omega$ times the value of that outcome $X(\omega)$? (To be clear, this is a *choice* that we make! We don't need to simply multiply the two quantities together. We could add them together, or square each of them and then multiply, or whatever we like. We *chose* to structure things this way). The answer is that the expected value was motivated via the real world concept of an average. Well, it turns out that multiplying our probability and our value together, and summing up the results across all possible outcomes, yields what we would get from a traditional average - and what we would get via running an experiment many times. Because probability was heavily designed from an *applied* vantage point (we wanted it to be able to solve real world problems), this property was ideal. And, it turns out that this property really just requires us to say: "we want a fraction of $x_1$ in our expected value, a fraction of $x_2$ in our expected value, and so on". It is in this way that we create a mathematical object that nicely captures our real world concept of average. ### Multiplication is a shortcut for *enumeration* and *counting* Multiplication is just a shortcut for enumeration and counting. Say we have the example of $7 \times 20$ again. This is equivalent to adding up $20$ seven times. In other words, splitting out seven groups of $20$ and counting them up. For more see Statistical Rethinking, pg. 25. --- Date: 20211011 Links to: [Expected Value](Expected%20Value.md) Tags: References: * [Multiplication as "of", Quora answer](https://qr.ae/TSyqIk) * [Multiplying non numbers](https://www.math3ma.com/blog/multiplying-non-numbers) * Statistical Rethinking, pg. 25