+Chapter 5Probability: What Are the Chances?

5.1 Randomness, Probability, and Simulation

5.2 Probability Rules

5.3 Conditional Probability and Independence

+Section 5.1Randomness, Probability, and Simulation

After this section, you should be able to…

DESCRIBE the idea of probability

DESCRIBE myths about randomness

DESIGN and PERFORM simulations

Learning Objectives

+Random

ness, Probability, and S

imulation

The Idea of Probability

Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run.

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.

+ 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

The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation.

Random

ness, Probability, and S

imulation

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.

Performing a Simulation

We can use physical devices, random numbers (e.g. Table D), and technology to perform simulations.

+ 1. State the problem or describe the random phenomenon Ex: Toss a coin 10 times, what is the likelihood of a run of at least 3

consecutive heads or 3 consecutive tails?

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)

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

+GOLDEN TICKET PARKING LOTTERY

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.

+

Example: Golden Ticket Parking Lottery

Read the example on page 290.

What is the probability that a fair lottery would result in two winners from the AP Statistics class?

Students Labels

AP Statistics Class 01-28

Other 29-95

Skip numbers from 96-00

Reading across row 139 in Table D, look at pairs of digits until you see two different labels from 01-95. Record whether or not both winners are members of the AP 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 40|00|36 00|24|28

✓ | ✓ Sk|X|X X | ✓ ✓ | X X | X X | X ✓ | ✓ X|Sk|X Sk|✓|✓

Yes No No No No No Yes No Yes

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.

+

EX: A breakfast cereal company decides to offer a 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.

+ Example: NASCAR Cards and Cereal Boxes

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.

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.

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

+

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 5.1Randomness, Probability, and Simulation

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.

Summary

+Looking Ahead…

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

In the next Section…