Finding probabilities

Welcome! Now, after you have acquired a solid basis, it is time for us to move on and learn more about types of probability models and calculate the probability of an event. In this topic, we will derive a formula for finding the general probability of an event, inspect a separate case of a sample space with equally likely outcomes, and learn about a distinction between discrete and continuous probability models.

Finding probabilities for a probability model

Let's begin with a classic example – we roll a single, 6-sided dice and we want to know the chances of us rolling 6. It is intuitive to assume that the probability, in this case, would be equal to ⅙, and you would be correct! There are only 6 different numbers on the dice and each one is equally likely to be rolled.

But how could this be written more formally? Let's see, after we roll the dice there are 6 possible outcomes: we get a number from 1 to 6. This is our sample space $\Omega$ = {1,2,3,4,5,6} – a set with 6 elements. Then we define another set A. It contains all outcomes that satisfy our condition. In this case, set A would contain only one element: A = {6}.

Now how could we find the probability of us rolling a 6? Let's rephrase it a bit: "How many cases out of all possible ones satisfy our condition?". We can rewrite this question using formulas. The number of cases that satisfy our condition corresponds to the number of elements in set A, and the number of all possible outcomes corresponds to the number of elements of the set $\Omega$, so we get the following expression: $|A|\over|Ω|$. Let's plug in the numbers and see what happens:

|A| = 1, |Ω| = 6, so ${|A|\over|Ω|} = {1\over6}$ — exactly what we had in the beginning, good job!

And just like that, we derived the main formula for finding the probability, where we divide the number of positive(desired) outcomes by the number of all possible outcomes.

Let's take a look at another example. We roll the same dice a single time once again, but now we want to know the probability of us rolling an even number. The sample space $\Omega$ stays the same as before but set A is different and contains the following elements: {2, 4, 6}. There are 3 elements in set A, so let's plug in that value into the formula:

${|A|\over|Ω|} = {3\over6} = {1\over2}$ so there is a 50% chance of us rolling an even number.

This is the fundamental formula that is used for calculating the probability of sample spaces. For now, we have dealt only with basic examples where it isn't hard to define A and find |A| and |$\Omega$|. This may not be the case in more complicated tasks, and we will learn special tricks and techniques on how to find those numbers in the future.

Finding probability for a sample space with equally likely outcomes

You might have noticed that in the examples above the probabilities of all possible events were equal – in the first one our sample space contained 6 events and the probability of any event was $1\over6$, and in second we had only 2 possible outcomes – either we will roll an even number or an odd one, so the probability of each of those was $1\over2$. Such sample spaces, where all events have the same probability, are called equiprobable sample spaces. It is very easy to find a probability of an event in such probability space. It only requires knowing the number of all possible events. The probability of an event would be one divided by the number of all possible events.

For example, let's imagine that your friend invented the time machine that could send you to the past. Unfortunately, the machine is imperfect as it can only send you to a particular year, but not a month or a day. But you would like to give it a shot. What is the probability of you traveling to the 1st of April of any non-leap year? Because you are equally likely to travel to any day in the year, this is an equiprobable sample space with 365 possible events, so the probability would be $1\over365$.

Discrete and continuous probabilities

If you remember, previously we have learned that sample spaces can be either countable or uncountable. In statistics, they are called discrete and continuous respectively. There is a big difference between them and how we compute the probabilities for them. In discrete spaces, we handle each event separately. In today's examples, we only worked with discrete spaces. In continuous spaces, the probabilities of all individual events approach 0 and we are unable to carry out calculations with separate events. Instead, we work with a range of events. Let's look at an example.

Imagine you are playing golf. Based on your skill, strength, and other factors you definitely know that after your strike, the ball will land somewhere inside a particular circle area. Inside this area there is a sand bunker of a square shape, if the ball gets there, it severely decreases the accuracy of your next shot, so you want to avoid it.

Whether you take the shot or not depends on the probability of you not hitting the bunker, so you want to find this probability. Because the ball will land at a completely random place inside the circle zone, the probability of us hitting the bunker would be equal to the ratio of bunker's and circle's areas: $S(bunker)\over S(circle)$. In this example, we are dealing with the discrete probability space because it had only two events (hitting and not hitting the bunker) and both we may treat separately to compute their probabilities.

But what if we would like to know what are the chances of the ball landing at a specific point inside the circle zone? From a geometric point of view, the circle is composed of an endless number of points, such that each individual point has no area. So, if we use the formula from the previous example, we will get a zero probability. In other words, the ball can't land in a specific spot inside the circle, which, obviously, is not true. The problem lies in our interpretation of events. Instead of separate points, we should look for collections of points or rather subareas inside our bigger circle area, for example, the bunker from the previous example. This time we dealt with the continuous probability — where we can't operate with individual events and instead do it with their collections.

Discrete probability spaces are much more common in everyday use. The probability of winning in a card game, passing an exam, or winning a lottery are all discrete probabilities. Continuous probabilities are mostly used for scientific purposes.

Conclusion

Well done! In this topic, we have learned about the formula for computing a general probability of an event (${events \space that \space satisfy \space the \space condition \over all \space possible \space events}$) and used it in practice. You have learned that all events in equiprobable sample spaces have the same probability. And finally, we made a distinction between discrete probability spaces, in which we treat each event as a separate entity, and continuous probability spaces, where we can't operate with singular events and instead do it with the ranges and collection of events.