Download A.P Statistics Lesson 6-2: Probability Models The sample space S of

A.P Statistics Lesson 6-2: Probability Models
The sample space S of a random phenomenon ____________________________________________________
An event is ________________________________________________________________________________
An event is a subset of the sample space.
A probability model is ______________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
A sample space s can be very simple or very complex. When we toss a coin once, there are only two outcomes,
heads and tails. The sample space is S = {H, T}. If we draw a random sample of 50,000 U.S. Households, the
sample space contains all possible choices of 50,000 of the 103 million households in the country. This S is
extremely large. Each member of S is a possible sample.
Example 1: Rolling two dice is a common way to lose money in casinos. There are 36 possible outcomes
when we roll two dice and record the up-faces in order (first die, second die). The figure below shows these
outcomes. They make up the sample space S.
“Roll a 5” is an event, call it A, that contains ______
of these 36 outcomes.
A={
}
Example 2: Let your pencil fall blindly into Table B of random digits; record the value of the digit it lands on.
The possible outcomes are:
S={
}
Example 3: An experiment consists of flipping a coin and rolling a die. Possible outcomes are a head (H)
followed by any of the digits 1 to 6, or a tail (T) followed by any of the digits 1 to 6. What are all of the
possible outcomes in this sample space?
There are two techniques that are very helpful in making sure you don’t accidently overlook any outcomes. The
first is called a tree diagram.
In Example 3 the first action is to toss a coin. To construct a tree diagram,
begin with a point and draw a line from the point to H and a second line
from the point to T. The second action is to roll a die; there are six possible
faces that can come up on the die. So draw a line from each of H and T to
these six outcomes.
The second technique is to make use of the following rule.
MULTIPLICATION PRINCIPLE (Fundamental Counting 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 • b number of ways.
In Example 3, there are 2 ways the coin can come up and there are 6 ways the die can come up, so there are
2 • 6 possible outcomes in the sample space.
Example 4: An experiment consists of flipping four coins. What is the total possible number of outcomes for
this sample space?
List all the possible outcomes of this sample space.
0 heads
1 head
2 heads
3 heads
4 heads
If you are selecting objects from a collection of distinct choices, such as drawing playing cards form a standard
deck of 52 cards, then much depends on whether each choice is exactly like the previous choice. If you are
selecting random digits by drawing numbered slips of paper from a hat, and you want all ten digits to be equally
likely to be selected each draw, then after you draw a digit and record it, you must put it back into the hat. Then
the second draw would be exactly like the first. This is referred to as sampling with replacement. If you do
not replace the slips you draw, there are only nine choices for the second slip picked, and eight for the third.
This is called sampling without replacement.
Example 5: How many three-digit numbers can you make using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9?
With replacement:
Without replacement: