Transcript Sec. 6.1 and 6.2 Part 1 PowerPoint

CHAPTER 6: PROBABILITY
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.
COIN TOSSING EXAMPLES

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
CHAPTER 6: PROBABILITY
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 subset 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 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 2 4 giving a total of 16 outcomes.
 The list can be organized in different ways:
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

Suppose our only interest in the last example is
the number of heads we get in four tosses. What
is the sample space?


Probability models can also be used to determine
possible outcomes for other situations.


S = {0, 1, 2, 3, 4}
For example, the possible outcomes of an SRS of 1500
people are the same as when flipping a coin 1500
times 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.

Homework:
Sec. 6.1: p.334 #4
 Sec. 6.2 part 1: p.340-342 #’s 11, 15, 17, & 18
