Download + Section 5.1 Randomness, Probability, and Simulation

+
Chapter 5
Probability: What Are the Chances?
 5.1
Randomness, Probability, and Simulation
 5.2
Probability Rules
 5.3
Conditional Probability and Independence
+ Section 5.1
Randomness, Probability, and Simulation
Learning Objectives
After this section, you should be able to…

DESCRIBE the idea of probability

DESCRIBE myths about randomness

DESIGN and PERFORM simulations
Idea of Probability
The law of large numbers says that if we observe more and more
repetitions of any chance process, the proportion of times that a
specific outcome occurs approaches a single value.
Definition:
The probability of any outcome of a chance process is a
number between 0 (never occurs) and 1(always occurs) that
describes the proportion of times the outcome would occur in a
very long series of repetitions.
Randomness, Probability, and Simulation
Chance behavior is unpredictable in the short run, but has a regular and
predictable pattern in the long run.
+
 The

Introduction:

Probability is the study of chance.

When we produce data by random sampling or randomized
comparative experiments laws of probability answer the
question “what would happen if we did this many times?”

Probability is the basis of inference
+
+
http://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.html
+

Simulation:
A couple plans to have children until they have a girl or until
they have 4 children, whichever comes first. What are the
chances that they will have a girl among their children?
 Let a flip of a fair coin represent a birth, heads = girls,
tails = boy (since both outcomes are equally likely the
coin is an accurate imitation of the situation)
 Flip the coin until a head appears or 4 times, whichever
comes first.
 If this coin flipping procedure is repeated many times,
then the proportion of times that a head appears within
the first 4 flips should be a good estimate of the true
likelihood of the couple’s having a girl.
 What’s another tool we could use to simulate birth
scenario?
+
 Simulation
Performing a Simulation
State: What is the question of interest about some chance process?
Plan: Describe how to use a chance device to imitate one repetition of the
process. Explain clearly how to identify the outcomes of the chance
process and what variable to measure.
Do: Perform many repetitions of the simulation.
Conclude: Use the results of your simulation to answer the question of
interest.
We can use physical devices, random numbers (e.g. Table D), and
technology to perform simulations.
Randomness, Probability, and Simulation
The imitation of chance behavior, based on a model that
accurately reflects the situation, is called a simulation.

+


1. State the problem or describe the random phenomenon

2. State the Assumptions (there are 2)

A head or a tail is equally likely to occur on each toss

Tosses are independent of each other
3. Assign digits to represent outcomes (want efficiency)



Ex: Toss a coin 10 times, what is the likelihood of a run of at least 3
consecutive heads or 3 consecutive tails?
In a random # table, even and odd digits occur with the same long-term
relative frequency (50%)

One digit simulates one toss of the coin

Odd digits represent heads; even digits represent tails
Successive digits in the table simulate independent tosses
4. Simulate many repetitions

Looking at 10 consecutive digits in Table B simulates one repetition.
Read many groups of 1- digits from the table to simulate many rep
At a local high school, 95 students have permission to park on campus.
Each month , the student council holds a “golden ticket parking lottery”
at a school assembly. Two lucky winners are given reserved parking
spots next to the main entrance. Last month, the winning tickets were
drawn by a student council member from the AP Statistics class.
When both golden tickets went to members of that same class, some
people thought the lottery had been rigged. There are 28 students in
the AP Stat class, all of whom are eligible to park on campus.
DESIGN AND CARRY OUT A SIMULATION TO DECIDE WHETHER IT
IS PLAUSIBLE THAT THE LOTERY WAS CARRIED OUT FAIRLY.
+
GOLDEN TICKET PARKING LOTTERY
Golden Ticket Parking Lottery
+
 Example:
Read the example on page 290.
What is the probability that a fair lottery would result in two
winners from the AP Statistics class?
Reading across row 139 in Table
Students
Labels
D, look at pairs of digits until you
AP Statistics Class 01-28
see two different labels from 0195. Record whether or not both
Other
29-95
winners are members of the AP
Skip numbers from 96-00
Statistics Class.
55 | 58
89 | 94
04 | 70
70 | 84
10|98|43
56 | 35
69 | 34
48 | 39
45 | 17
X|X
X|X
✓|X
X|X
✓|Sk|X
X|X
X|X
X|X
X|✓
No
No
No
No
No
No
No
No
No
19 | 12
97|51|32
58 | 13
04 | 84
51 | 44
72 | 32
18 | 19
✓|✓
Sk|X|X
X|✓
✓|X
X|X
X|X
✓|✓
X|Sk|X
Sk|✓|✓
Yes
No
No
No
No
No
Yes
No
Yes
40|00|36 00|24|28
Based on 18 repetitions of our simulation, both winners came from the AP Statistics
class 3 times, so the probability is estimated as 16.67%.
So after 18 repetitions , there have been 3 times when both
winners were in the AP Stat class.

If we keep going for 32 more repetitions ( to bring our total to
50) , we find 30 more “no” and 2 more “ yes” results. All totaled
that’s 5 “ Yes” and 45 “ No” results.
Conclude: In our simulation of a fair lottery, both winners came
from the AP Stat class in 5/ 50 = 10% of the repetitions.
+

NASCAR promotion. Each box of cereal will contain a
collectible card with NASCAR drivers:
Jeff Gordon, Dale Earnhardt,Jr., Tony Stewart, Danica
Patrick, or Jimmie Johnson.
The company says that each of the card is equally likely
to apper in any box of cereal. A NASCAR fan decides to
keep on buying boxes until she has all 5 drivers’ cards.
She is surprised when it takes her 23 boxes to get the
full set of cards. Should she be surprised?
Design and carry out a simulation to help answer this
question.
+
EX: A breakfast cereal company decides to offer a
NASCAR Cards and Cereal Boxes
+
 Example:
Read the example on page 291.
What is the probability that it will take 23 or more boxes to get a
full set of 5 NASCAR collectible cards?
Driver
Label
Jeff Gordon
1
Dale Earnhardt, Jr.
2
Tony Stewart
3
Danica Patrick
4
Jimmie Johnson
5
Use randInt(1,5) to simulate buying one box of
cereal and looking at which card is inside. Keep
pressing Enter until we get all five of the labels
from 1 to 5. Record the number of boxes we
had to open.
3 5 2 1 5 2 3 5 4 9 boxes
4 3 5 3 5 1 1 1 5 3 1 5 4 5 2 15 boxes
5 5 5 2 4 1 2 1 5 3 10 boxes
We never had to buy more than 22 boxes to get the full set of cards in 50 repetitions of
our simulation. Our estimate of the probability that it takes 23 or more boxes to get a
full set is roughly 0.
In the Golden ticket lottery example, we ignored repeated numbers
from 01 to 95 within a given repetition. That’s because the chance
process involved sampling students without replacement.

In the Nascar example, we allowed repeated numbers from 1 to 5 in a
given repetition. That’ s because the chance process of pretending to
buy boxes of cereal and looking inside could have resulted in the
same driver’s card appearing in more than one box.
+

+ Section 5.1
Randomness, Probability, and Simulation
Summary
In this section, we learned that…

A chance process has outcomes that we cannot predict but have a
regular distribution in many distributions.

The law of large numbers says the proportion of times that a
particular outcome occurs in many repetitions will approach a single
number.

The long-term relative frequency of a chance outcome is its
probability between 0 (never occurs) and 1 (always occurs).

Short-run regularity and the law of averages are myths of probability.

A simulation is an imitation of chance behavior.
+
Looking Ahead…
In the next Section…
We’ll learn how to calculate probabilities using
probability rules.
We’ll learn about
 Probability models
 Basic rules of probability
 Two-way tables and probability
 Venn diagrams and probability