```Section 6.1 – The Idea of Probability
The Idea of Probability

Probability begins with the observed fact that some phenomena are random – that is, the relative frequencies of
their outcomes seem to settle down to fixed values in the long run.

The big idea is this: chance behavior is unpredictable in the short run but has a regular and predictable pattern
in the long run.

The tossing of a coin cannot be predicted in just a few flips, but there is a regular pattern in the results, a pattern
that emerges clearly only after many repetitions.
Example 6.1 on p.331


For the first few tosses the proportion of heads fluctuates quite a bit, but settles down as we make more
and more tosses.
Also read example 6.2 on p.332
Section 6.2 Part 1 – Probability Models
Basic Descriptions of Probability Models:

A probability model is a mathematical description of a random phenomenon consisting of two parts:

A set of all possible outcomes, the sample space S.

A way of assigning probabilities to events.

A sample space can be very simple (such as rolling a single die) or very complex (rolling 100 dice).

An event is any outcome or a set of outcomes of a random phenomenon.


An event is essentially a ________________ of a sample space.
Example:

Consider tossing two fair coins:

The sample space (or list of all outcomes) would be S = {HH, HT, TH, TT}

Examples of events could include: P(0 heads), P(1 head), P(2 heads), or P(HT)

The probabilities for each event are based on the fact that there are four possible outcomes
Example 6.3 – Rolling Two Dice

Rolling two dice will have a total of 36 outcomes:

If you were to roll the two dice and record the up-faces in order (first die, second die) you would have a sample
space consisting of all 36 outcomes shown above.

If you were to change the scenario to only caring about the number of pips on the up-faces of the dice then the
sample space would be:

S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

“Rolling a sum of 5” is considered an event, which can occur multiple ways:

P(sum 5) = 4/36 or 1/9
Example 6.5 – Flip a Coin and Roll a Die

An experiment consists of flipping a coin and rolling a die. Find the sample space.

Use a tree diagram to represent the possible outcomes:
Multiplication Principle

If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done
in a x b number of ways.

Example:

We already found that there are 12 outcomes when flipping a coin and rolling a die. To
determine the number of outcomes without the tree diagram take the number of outcomes for
flipping a coin (2) and multiply it by the number of outcomes for rolling a die (6).
Example 6.6 – Flip Four Coins

Flipping four coins. What is the size of the sample space? List all possibilities.

The total number of outcomes (or size of the sample space) can be represented by 2 × 2 × 2 × 2 or 24
giving a total of 16 outcomes.

The list can be organized in different ways:

Suppose our only interest in the last example is the number of heads we get in four tosses. What is the sample
space?


S = {0, 1, 2, 3, 4}
Probability models can also be used to determine possible outcomes for other situations.

For example, the possible outcomes of an SRS of 1500 people are the same as
___________________________________________ when a question is answered “yes” or “no”.

Some sample spaces are simply too large to be able to list all possible outcomes in which case computer
programs are used.
Additional Definitions

With replacement refers to the drawing of an object, recording the selection and then putting it back so that it
may be drawn again.

Example:


How many 3 digit numbers can you make?

10 × 10 × 10 = 1000

This is assuming that each digit is eligible for each of the 3 positions
Without replacement refers to the drawing of an object, recording the selection and then not putting it back so
it can’t be drawn again.

Example:

How many 3 digit numbers can be written if the numbers cannot be repeated?

10 × 9 × 8 = 720
```