Rebekah Robinson, Georgetown College

Always remember to make sure the necessary packages are loaded:

```
require(mosaic)
require(tigerstats)
```

Describe populations and samples using probability language.

Compute probabilities.

Probability is a way to quantify the *likelihood* that an event occurs. The probability of an event is a number between 0 and 1.

If the probability of an event is based on a personal belief that the event will occur, it is called a

subjective probability.

**Example**: A surgeon performing her very first surgery tells her patient that she feels the probability that the surgery will be successful is 0.99.

If the probability of an event is based on reasoning or calculation, it is called a

theoretical probability.

**Example**: If a fair coin is tossed, the *theoretical probability* that it will land heads is ½.

The

sample spaceof an experiment is the set of all possible outcomes.An

eventis some collection of these outcomes.

If all outcomes of an experiment have the same likelihood of occurring, then the probability of an event is

\[ P(event)=\dfrac{\mbox{# outcomes in the event}}{\mbox{# outcomes in the sample space}}. \]

**Example:** A *fair* coin is tossed **twice**.

Sample Space of Experiment = {HH, HT, TH, TT}

Probability of tossing 2 heads = \( P(HH)=\frac{1}{4}=0.25 \).

Probability of tossing 1 head = \( P(HT, TH)=\frac{2}{4}=0.5 \).

Probability of tossing 0 heads = \( P(TT)=\frac{1}{4}=0.25 \).

If the probability of an event comes from knowing the proportion of times the event occurs when the experiment is performed many times, it is called a

long-run frequency probability.

**Example:** A coin that is tossed 1000 times lands heads up 502 times. The *long-run frequency* is \( \frac{502}{1000}=0.502 \).

The *long-run frequency* provides a good **approximation** to the *theoretical probability* when the experiment is performed many times.

A

random variableis a variable whose value is the outcome of an experiment.

Before an experiment, a random variable

- is full of potential.
- can take on any value in its range.
- may be more likely to take on some values than others.
- is denoted with uppercase letters \( X, Y \), or \( Z \).

After an experiment, a random variable

- is known.
- is denoted using lowercase letters \( x,y \), or \( z \).

A

discrete random variableis a random variable whose possible values are whole numbers.

**Example:** A fair coin is tossed twice. One possible random variable associated with this experiment is

\[ X= \mbox{ number of heads tossed.} \]

Sample Space of Experiment = {HH, HT, TH, TT}.

Range of Random Variable = {0, 1, 2}.

Some values for \( X \) are more likely than others.

\( P(X=0) = P(TT) = \frac{1}{4} = 0.25 \).

\( P(X=1) = P(HT, TH) = \frac{2}{4} = 0.5 \).

\( P(X=2) = P(HH) = \frac{1}{4} = 0.25 \).

The

probability distribution function (pdf), \( P(X=x) \), for a discrete random variable \( X \) is a function that assigns probabilities to all of the values in the range of \( X \).

The **pdf** can be viewed as a table…

\( x \) | 0 | 1 | 2 |
---|---|---|---|

\( P(X=x) \) | 0.25 | 0.50 | 0.25 |

… or a histogram.

\( x \) | 0 | 1 |
2 |
---|---|---|---|

\( P(X=x) \) | 0.25 | 0.50 |
0.25 |

\[ P(X=1)=0.5 \]

\( x \) | 0 | 1 |
2 |
---|---|---|---|

\( P(X=x) \) | 0.25 | 0.50 |
0.25 |

\[ P(X\geq 1)=0.75 \]

Part 2 will discuss **expectation**, **standard deviation**, and a special discrete random variable.