Prove that set of all rational numbers is countable

This claim is proven by building a bijection by following a diagonal procedure.

Imagine you have an infinite matrix, whose coefficient are all possible fractions:

Now, count them "diagonally": this is how we map every natural number, using the same positions:

The attached image, taken from aminsaied.wordpress.com/2012/05/21/diagonal-arguments/, should make it clearer

0.16 or 16%

step-by-step explanation:

lets take a coin for example.

if you toss it twice, it has a 1/4 chance of landing on heads both times because 1/2 * 1/2 is 1/4.

with the cards, the chance of picking dog is 0.4. having two people pick one is like picking a card twice in a row. so 0.4 * 0.4 is 0.16, or 16%