Sample Space, Outcomes, Events in Probability

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In Probability Theory, we work on measuring how likely something is to happen. Whether we are flipping a coin, rolling a dice, or predicting the weather, Probability helps us find out the chances.

Probability Theory covers some fundamental concepts like experiments, outcomes, sample spaces, and events. In this chapter, we will explain these concepts one by one and see how they are important. We will take a couple of examples, like flipping coins, rolling a dice, or even something as random as how much rain falls in a year, to help you understand the concepts better.

What is an Experiment in Probability Theory

In Probability Theory, an experiment is simply an action or process that has observable result. This result could be anything from flipping a coin, rolling dice, or, in more complex cases. Each time we perform this action, we are doing a trial. For example, one flip of a coin is one trial, and the result could be heads or tails.

Example of Coin Flip − Consider an experiment where we flip a coin. The outcome of the coin flip could either be heads or tails. It is useful because it shows how every trial gives us one of the two possible outcomes: heads or tails.

Example of Running a Maze − Another interesting experiment is like a rat running through a maze. Let us say there are three possible paths the rat can take. Every time the rat runs the maze, this is a trial. One possible outcome would be that the rat takes path number one.

Example of Rainfall − A little complex example could be measuring the rainfall in a city over one year. So, the trial in this case is observing the rainfall over one year, and the outcome is some specific number, like 37.23 inches of rain.

Sample Space: The Complete Set of Outcomes

In a sample space, we look at all the possible outcomes that can come from trials. This complete set of outcomes is called the sample space.

We can think of it as the "universal set" in set theory. Every possible result of the experiment is part of this set.

Example: Throwing Two Dice

Let us see the same example again. We roll two dice, one red and one green. The sample space for this experiment would include every possible combination of the numbers that show up on the dice. So, if we roll the red die and get a 2, and the green die get 5, one possible outcome would be (2, 5).

There could be a confusion. We might think the sample space for rolling two dice would be just the numbers 2 through 12 (The sum of the dice can be), this is not right. Each pair of numbers on the dice, like (2, 5) or (5, 2), is a unique outcome. So, instead of focusing on the sum, we need to look at each individual pair.

For two regular dice, the sample space would actually have 36 possible outcomes (since each die has 6 sides, and 6 times 6 equals 36). Every pair, like (1, 1), (1, 2), all the way to (6, 6), and that will be the sample space.

Events: Subsets of the Sample Space

An event is nothing but a subset of the sample space. That says one or more of the possible outcomes that we care about. When we say an event occurs, we mean is that the outcome of the experiment is in that subset.

Example: Sum Greater Than Nine

Let us take an example with dice. Imagine we are finding an event where the sum of the two dice is greater than nine. Now, for this event, we are looking for pairs where the numbers add up to 10, 11, or 12. Those pairs would be −

• (4, 6)
• (5, 5)
• (5, 6)
• (6, 4)
• (6, 5)
• (6, 6)

So, this event, "sum greater than nine," is a subset of the full sample space of 36 outcomes. It contains just these six outcomes.

Example: Sum Equals Seven or Eleven

Let us see another similar example. What if it will be if rolling a sum of seven or eleven? For this event, the outcomes would be −

• For seven: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
• For eleven: (5, 6) and (6, 5)

In this case, there are 8 outcomes in our event.

Special Events: The Impossible and Certain Events

Let us see two extreme kinds of events: the impossible event and the certain event.

• The Impossible Event − The impossible event is like trying to get a student with a birthday on February 30^th, there is no such thing since February never has 30 days. The impossible event corresponds to the empty set. This means it has no outcomes in it.
• The Certain Event − On the other side, we get the certain event. This is something that will definitely happen. In our dice example, a certain event would be that the dice show some number between 1 and 6. There is no way to roll anything else. So this event corresponds to the entire sample space.

Events as Sets: Unions, Intersections, and Complements

In probability, events can be treated just like sets, which means we can combine them using set operations like union, intersection, and complement.

Union of Events

The union of two events, can be written as E ∪ F. Either event happens, or both happen. So, for example, if E1 is the event where the sum of two dice is greater than nine, and E3 is the event where the two dice show the same number, E1 ∪ E3 would include all outcomes where either of these things happens.

Intersection of Events

The intersection, written as E ∩ F, is where both events happen at the same time. Means if looking at E1, where the sum is greater than nine, and E3, where the dice are the same number, E1 ∩ E3 would include just the outcomes where both conditions are met. In this case, it would be (5, 5) and (6, 6), since those pairs are both greater than nine and show equal numbers.

Complement of an Event

The complement of an event is written as E^c. This includes everything in the sample space which not in the event. So, if E1 is the event where the sum is greater than nine, E1^c would be all the outcomes where the sum is nine or less.

Mutually Exclusive Events

Two events are mutually exclusive if they cannot happen at the same time. For example, if E2 is the event where the sum is seven or eleven, and E3 is the event where the two dice are equal, these two events are mutually exclusive. The dice will not add up to seven or eleven if both dice show the same number, so there are no outcomes that belong to both events.

Conclusion

In this chapter, we presented the basic concepts of probability related to experiments, outcomes, sample spaces, and events. We also understood how events can be combined using unions, intersections, and complements, and explained what it means for events to be mutually exclusive.