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Probability Notation
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Probability is the likelihood of an outcome. Before we can properly define probability, we must first define 'events.' It is helpful and convenient to denote the collection of events as a single letter rather than list all possible outcomes.

- Event
- a collection of outcomes, typically denoted by capital letters such as A, B, C, etc...

#### Examples of Events

- Suppose we ask 30 students to record their eye color. We can define an event B to be blue eye color. In other words, let $B=\{\text{blue eyes}\}$.
- A game is played where you roll two fair six-sided die. A player is allowed to “roll again” if there are doubles, (i.e. both die show the same face). Define the event R as a collection of outcomes that allow a player to roll again. Therefore, R={both 1’s, both 2’s, both 3’s, both 4’s, both 5’s, both 6’s}.

Often students in introductory statistics courses struggle with probability due to getting caught up and/or confused with the general notation used in describing events and the associated probability. To put it simply, the notation is shorthand to keep one from continually needing to write out long phrases to explain what is taking place. For instance, if one were to consider the toss of a fair coin the common theme is that there is a 1/2 chance, or 0.5 probability, of the coin coming up Tails. But how does one write this event?

**Converting to Probability Notation**

- Identifying the outcome event of interest: {Getting a Tail when we toss a fair coin}.
- Use a single letter or word to represent this outcome of interest: T={Getting a Tail when we toss a fair coin}, for instance.
- State your interest in the probability of this outcome: P(T) which is read, "Probability of getting a Tail when we toss a fair coin."

When you read a statistics text book a common lettering system uses the beginning of the alphabet. That is, the authors use 'A', 'B', etc. to define outcome events of interest.

As you can see, the lettering can become convoluted! Just remember that the key is to identify what your outcome event of interest is.

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Try It! Probability Statements
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Example 2-1
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Now let's complicate things. Consider again tossing a fair coin. We stated P(T) was the probability we get a tail when we toss the coin. How would one write the probability statement if the outcome was getting two tails when the coin was tossed twice?

#### Answer

Applying the steps we get...

- Identify the outcome of the event:
*Getting a tail on both tosses of a fair coin.* - Use a single letter or word to represent this outcome of interest:
*We can write this as T,T, where the first T represents the outcome of the first toss and the second T as the outcome of the second toss.* - State your interest in the probability of this outcome:
*P(T,T) which is read, "Probability of getting a Tail on the first and second toss."*

*Note: Often the comma is eliminated and P(T,T) is written as P(TT).