How Einstein Proved that atoms existed. And How you will too. Part 2.1
Probability theory helps us find patterns in the world when things are not predictable. Our first example will be a deck of cards, where cards are being drawn at random, which is an example of a discrete probability distribution. An example of a continuous probability distribution is betting on the finish time of a race.
Let’s do our first calculation, this one should be easy. In a deck of 52 cards, what is the probability of picking the ace of spades? What is the probability of picking any ace? What is the probability of picking any spade card? Write down your answers on a piece of paper, or better yet post it in the comments!! it will help you remember what you learn. After attempting to answer, check the video below.
Second calculation: what is the set of possible outcomes for a coin toss? what about a double coin toss (two coin flips)? What is the likelihood of two coins having the same result.
Again, write down your answer, before continuing to the video. The game below demonstrates to you coin tossing and probability, Enjoy!
Probability theory basically assigns a number between 0 and 1 to each event. For example the probability of a spade card is 1/4, the probability of two coins being different is 1/2 and so on. These number can’t be anything, there are two main restrictions on them. Firstly, if the event is the entire collection of outcomes then the probability must be 1. That’s because you are sure that whatever the outcome is, it will be in the set of all possible outcomes!! Secondly if you combine two mutually exclusive events, events that don’t share any outcomes, then the probability of the combined event is the sum of the individual probabilities. We already saw this is in the first example, we added the probabilities of individual spade cards to get the combined probability of a spades event.
Third calculation: In a game of pool you have to pick up the balls and place them on top of the table, There are 15 balls in the hole to begin with, this calculation is about the first 5 balls you pick. How many different ordered combination are there? For example 12345 is on of them, 34521 is another because you care about the order and so is 68135. (Hint: start by thinking about how you would do this if you where picking two balls from just the 123 set, then generalize to the current problem).
This calculation is related to probability because, once you know the probability of the individual outcomes, (discrete) probability theory is just counting how many outcomes are in a given event.
Fourth calculation: Repeat the third calculation for the case where you don’t care about the order you picked the balls, you just grab them all at once, so that 12345 and 54321 counts as the same event.
Next post we will understand averages and standard deviation.