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Data records what has happened. Probability is a branch of statistics that investigates the chances of what will happen in the future. Probabilities, colloquially also referred to as chances, are systematic attempts to determine how likely particular events are to occur, so for example you have a 1 in 6 chance or probability for a six to show up when you roll a fair dice. Sometimes, however, you cannot be that precise, for example when you say that something is "likely," "unlikely," or "almost certainly," happening, you are implicitly giving an imprecise and rough guess about the actual probability of the event, but you are still applying probabilities to do that. Probabilities help you to understand what data you will record in the future.

This lesson will help you comprehend the fundamentals of probability and chance, so you can better understand what data you will see.

Introduction to Probability and Chance

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Random Experiment

Until now you only looked at data, may be considered the sampling process you used to collect it, but have not yet explored its origin and the process underlying its realization. As you are now moving from descriptive statistics to inferential statistics, which concerns itself with how to draw conclusions from a sample about the underlying population, you will now look more closely into uncertainty and the probability of certain outcomes or events occurring.

Here are some examples of random experiments and their respective outcomes

Notice, in all case you do know what the possible outcomes of the process is, so you know that flipping a coin will result in either head or tail, but you do NOT know which one will show up if you do. While you do know what can occur, it is uncertain which actual outcome does happen. When you review the previous modules about data, think about what the underlying random experiment is and what the uncertain outcomes are that can occur.

Simple Events and Sample Space

When you have a random experiment, you also have the potential for different outcomes, which are also called "events". For now focus on the most simple, most basic events. So for example, for rolling a die once you could take "roll a number higher than 3" as an event, but it's not the most simple one. "Rolling a 1", however, is such a simple event.

A list of all possible simple events (or outcomes) of a particular random experiment is called a sample space.

Formally, the sample $S$ consists of the different simple events $O_i$, so $S = \left\{O_1, O_2, …,O_k\right\}$

What does this practically look like?

The sample space of rolling a die once is a number between 1 and 6, so $S = \left\{1, 2, 3, 4, 5, 6\right\}$ and for the rest of the examples from above.

Practice: Sample spaces

Imagine you have 4 tickets to a business conference and you can only take 3 of your teammates. You can choose from Abby (A), Brian (B), Chloe (C), and David (D). How many different choices of groups of colleagues do you have?

Directions: Take a piece of paper and write down the list of all possible arrangements for your three colleagues. Then cross out any duplicate groupings that represent the same group of friends.

All possible arrangements: (the following four lines of letter combinations should be accordion style shown when the student clicks on arrangements)

ABC ACB BAC BCA CAB CBA

ABD ABD BAD BDA DAB DBA

ACD ADC CAD CDA DAC DCA

BCD BDC CBD CDB DBC DCB

Arrangements after crossing out duplications (the following four lines of letter combinations (including crossed out ones) should be accordion style shown when a student clicks on arrangements)

ABC ACB BAC BCA CAB CBA

ABD ABD BAD BDA DAB DBA

ACD ADC CAD CDA DAC DCA

BCD BDC CBD CDB DBC DCB

By listing all possible arrangements and then crossing out the duplicates you just created a list of simple events and the overall sample space. These combinations of three colleagues are the simple events for the random experiment of selecting three colleagues. And you determined the full sample space as you collected all possible arrangements, so the set is exhaustive as no other combination is feasible, and the simple events are mutually exclusive, if you pick three colleagues you cannot pick a different combination of them at the same time.

Two more questions to think about on your own:

• Can you think of a way to calculate the number of simple events when you have to distribute the three tickets over five (instead of four) colleagues?

• What are the sample space and the list of simple events if your three tickets were numbered (think about sitting at three different tables) and it makes a difference which of your four colleagues gets which ticket?

Probability of an event

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Florian is to do the explainer video in Napier with a separate intro video in Barr Smith

Probability of events

The main intention here is to find $P(A)$, the probability that event $A$ will occur.

An event is any collection of one or more possible simple events of an underlying random experiment, so the event $A$ is a collection of simple events $\left\{O_e, O_g, …,O_h\right\}$ from the sample space $S = \left\{O_1, O_2, …,O_k\right\}$ of the random experiment

The probability of an event, so the probability $P(A)$ of event $A$, is the \textbf{sum of the probabilities assigned to the simple events contained in} {\boldmath $A$}.

Focusing on the probability {\boldmath $P(O_i)$} of a simple event $O_i$, the following two characteristics for the probabilities of simple events contained in a sample pace must hold:

• $0 \leq P\left(O_i\right) \leq 1$, for all $i$

• $\sum\limits_{i=1}^{n} P\left(O_i\right)=1$

Where a probability of 0 means that an event cannot occur, and a probability of 1 means that the event occurs with absolute certainty.

Now, think about how the two characteristics of simple events in a sample space, namely that they are mutually exclusive and exhaustive correspond to these characteristics of their probabilities.

Being exhaustive means that there is no other outcome, no other simple event, that can occur, so clearly, the combined probability of all simple events in the sample space must add up to one. (because if it does not, it means there is another outcome that is not in the simple space that has a probability, a chance, of occurring.

Being mutually exclusive means that no two simple events can occur at the same time. If the combined probability of all events in the sample space were more than 1, then two events must occur at the same time (otherwise it is not possible to get above 1, or 100%). But as simple events are defined as mutually exclusive, so cannot happen at the same time, the total sum of the probability of all simple events cannot be above 1.