Because there are no fractional array elements

const array = [123, 456];
console.log(array[0.5]);

The code above prints undefined because there is no 0.5 element. There is a 0 element and a 1 element but no 0.5 element

So, to choose a random element with Math.random() which returns a number between 0 and 1 (but not 1) you have to convert to an integer.

JavaScript arrays are not like arrays in most other languages. They're actually more like augmented objects. They act like arrays in that you can get the length (array.length) and call things like push,pop,concat,shift,unshift but you can also add properties

array = [4, 5, 6];
array["foo"] = "bar"
console.log(Object.keys(array));

Prints

[
  "0",
  "1",
  "2",
  "foo"
]

Which explains why the Math.random without the Math.floor would fail. From the semantic point of view of JavaScript an array is just an object with properties named 0, 1, 2, 3 etc. Of course under the hood that's not what JavaScript does as it would be too slow to implement arrays that way but for the most part arrays act like they are actually just an object with properties named '0', '1', '2' and it's perfectly valid to have properties named '0.00012' etc.

Note, it would actually be faster to do this

const rand = array[Math.random() * array.length | 0];

The | 0 is a binary or of 0 which the JavaScript spec effectively says the result is converted to an integer before the | happens. Note that | 0 is not the same as Math.floor. | 0 rounds toward 0 whereas Math.floor rounds down.

         | 0   Math.floor         
------+------+------------
  2.5 |   2  |   2
  1.5 |   1  |   1
  0.5 |   0  |   0
 -0.5 |   0  |  -1
 -1.5 |  =1  |  -2
 -2.5 |  -2  |  -3

Math.floor() is preferred here because of the range of Math.random().

For instance, Math.random() * 10 gives a range of [0, 10). Using Math.floor() you will never get to the value of 10, whereas Math.ceil() may give 0.

Yes, this is indeed possible. Here's some example code:

Copy<script>
    var arr = new Array('a', 'b', 'c', 'd', 'e');
    document.write("Test " + arr[Math.floor(Math.random() * ((arr.length - 1) - 0 + 1))]);
</script>

Note that Math.floor should be used instead of Math.round, in order to get a uniform number distribution.

Why would anybody call Math.floor on a Math.random result?

In a nutshell, one calls Math.floor() when you want to truncate a decimal value to its nearest integer (by just dropping the decimal portion. So, 3.9 becomes 3, 2.1 becomes 2, etc... So, you would typically use that when you need an integer and you want the integer that is smaller than or equal to the decimal value. The math library also has Math.ceil() and Math.round(). Math.ceil() gets you the next larger integer, Math.round() rounds to the nearest integer going either up or down depending upon which is closer.

I've seen it used like: Math.floor(Math.random() * num);

Breaking Math.floor(Math.Random() * num) down into it's individual pieces and explaining each piece, you get this:

Math.random() gives you a random decimal number between 0 and 1, including 0, but not including 1. So, it might give you something like 0.38548569372.

Math.random() * num gives you a random decimal number between 0 and num, including 0, but not including num. So, if num was 10, it might give you 3.8548569372.

Math.floor(Math.random() * num)) gives you a random integer number between 0 and num, including 0, but not including num. So, it might give you 3.

Math.floor() truncates the decimal number to only the integer portion. A random integer is often used for getting a random value from an array (which needs to be an integer).

It seems a random color is desired - one with each component random between 0 and 255.

Math.random() returns a random number on [0,1) (ie it may be exactly zero or up to but not including one).

Multiplying that random value by 256 gives a random number on the range [0,256) (ie it may be 255.99, but never 256). Almost there, but not quite.

Math.floor() rounds the number downwards to the nearest whole integer, making the result an integer on [0,255] as desired.