Life or Lottery?  Determining the Probability

 

Course Description

  • Algebra II
  • Regular or Honors Curriculum
  • One 50-minute period
  • Fifth six weeks, week six, day 1

Overview

Teaching mathematics within a science framework can be both the motivating and the informative keys to delivering a curriculum. In particular, students are naturally inquisitive about space science and the topics surrounding the existence of intelligent life in other parts of our Universe. Tapping into this curiosity, this lesson uniquely combines the concepts of astronomy and probability in order for students to compare the likelihood of intelligent life existing elsewhere in the Universe and winning the lottery.

TEKS addressed:

B1 (B) in solving problems, the student collects data and records results, organizes the data, makes scatterplots, fits the curves to the appropriate parent function, interprets the results, and proceeds to model, predict, and make decisions and critical judgments. (Algebra II)

C4 (A) compare theoretical and empirical probability; (Mathematical Models with Applications)

Objectives

  • Students will use inquiry, problem solving, reasoning, and communication skills to compare the likelihood of intelligent life existing elsewhere in the Universe and winning the lottery.

Prerequisites

  • Students should have a basic understanding of probability and space science.

Materials

  • Overhead of Activity sheets 1-B and 3 (see below)
  • Program #12 in the Carl Sagan "Cosmos" series, Imagine the Universe! Web site or CD-Rom.
  • Decahedron dice (3 per group of three students)

Teacher Does                                     Probing Questions                              Student Does

Engagement

 “This question has been a focus of philosophical speculation since humans first contemplated the cosmos. Within recent decades, it has also become a topic of legitimate scientific inquiry within the field of astro-biology, the study of the origin, evolution, and distribution of life in the Universe. Current knowledge of the origin and nature of life, the process of the formation of stars and planets, and the evolution of intelligence and technology leads many scientists to speculate that there are millions of other potential "life sites", even within the Milky Way galaxy.”

“Scientists believe that the Universe was created about 15 billion years ago in a single violent event known as the Big Bang. All the space, time, energy, and matter that make up today's Universe originated in the Big Bang. The early Universe was extremely small, dense, and hot; it did not have a perfectly even distribution of energy and particles. These irregularities allowed forces to start to collect and concentrate matter. These concentrations of matter formed into clouds, and then condensed into stars and galaxies.”

“From the standpoint of the development of life, what matters is that each galaxy is a stellar factory, producing stars out of giant gas clouds. And each star is a chemical factory, transforming simple elements into heavier, more complex ones. Life is a collection of some of these complex molecules.”

 

 


”Now, before we go further with the discussion of life existing elsewhere in the Universe, we will establish some understandings about the lottery so that we can compare the likelihood of intelligent life existing elsewhere in the Universe and winning the lottery at the end of our lesson.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Explanation

Activity 1/ Part B
Next, we look at a lottery game which is slightly more complicated than Pick 3. This time we will examine the theoretical probability of winning a multiple digit game such as "Lotto".

If we calculate,

W=a * b * c * d * e * f * g


where,

W = The probability of winning the "Lotto"
a = The probability of 'getting' the first number
b = The probability of 'getting' the second number
c = The probability of 'getting' the third number
d = The probability of 'getting' the fourth number
e = The probability of 'getting' the fifth number
f = The probability of 'getting' the sixth number
g = The probability of 'getting' the seventh number

If we know that each number is one of forty and a given number cannot be called twice, we substitute

W=a * b * c * d * e * f * g


W= (1/40) (1/39) (1/38) (1/37) (1/36) (1/35) (1/34)

W= 1/93,963,542,400

an extremely small number!

 

 

 

 

Let's go back to thinking about the chances of intelligent life existing in the elsewhere in the Universe. You'll acquire some more background information by viewing a segment from Program #12 in the Carl Sagan "Cosmos" series, where this topic is discussed by the famous astronomer.

Viewing Activity: Program #12 in the Carl Sagan "Cosmos" series

  1. START the video 90 minutes into the tape at the point where you see and hear the Arecibo Observatory being discussed.
  2. PAUSE when you see Carl Sagan begin to explain the Drake Equation.
  3. FOCUS: Ask students to pay close attention to each value substituted in the Drake Equation. RESUME.
  4. PAUSE when Carl Sagan has solved the equation and obtained an answer.

 

 

 

 

6. STOP when Carl Sagan says "...enormously older and wiser than we"."

 

 

 

While working at the National Radio Astronomy Observatory in Green Bank, West Virginia, Dr. Frank Drake conceived a means to mathematically estimate the number of worlds that might harbor beings with technology that could communicate across the vastness of interstellar space. The Drake Equation, as it came to be known, was formulated in 1961 and is generally accepted by the scientific community.

N = fs * fp * ne * fld * fi * fc * fl

where,

N = The number of communicative civilizations in the Milky Way
fs = The number of stars in the Milky Way
fp = The fraction of those stars with planets (Current evidence indicates that planetary systems may be common for stars like the Sun.)
ne = The number of Earth-like worlds per planetary system
fld = The fraction of those Earth-like planets where life actually develops
fi = The fraction of life sites where intelligence develops
fc = The fraction of communicative planets (those on which electromagnetic communications technology develops)
fl = The fraction of a planet's lifetime that has a technological civilization

If we substitute,

N = fs * fp * ne * fld * fi * fc * fl


N = 400 billion (1/4) 2 (1/2) (1/10) (1/10) (1/100 million)


N ~ 10 technological civilizations in just the Milky Way Galaxy!

 

 

Multiply this by billions; the number of galaxies in the Universe!

Closure

Using transparencies of the two calculations (lottery and the Drake Equation) we are prepared to answer the following question;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Other assessments could include creating other lottery games where players are 'guaranteed' to win (modeling the likelihood of intelligent life existing elsewhere in the Universe).

Extensions

Students will include a probability connection in their moon projects.  Work should be included in their journals and results considered when finalizing their plan.  Other possibilities include:

  1. Research the number of Sun-like solar systems discovered thus far.
  2. Invite a lottery official to your classroom to discuss the state/multi-state lottery games.
  3. Have the science teachers in your school discuss the probability of inheriting certain characteristics (as modeled by the Punnett Square).
  4. Language Arts teachers can lead an activity on the probability of certain letters appearing in text. The same activity could be utilized by the Foreign language teacher, but of course this time examining the likelihood of letters appearing in text of a different language.

 

 

"Are we alone?"

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If we chose a certain series of 7 numbers, each less than 40 as is allowed in "Lotto", what would be our chances of winning?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.  FOCUS: Say: What would happen if we changed the value of fl? (The final answer will change. If fl is greater, the number of life forms existing is greater, vice versa.) RESUME.

 

 

 

 

Is there a way to estimate the number of technologically advanced civilizations that might exist in our Galaxy?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What would we do to determine the number of technological civilizations that exists in the Universe?

 

 

 

 

"Is there a better chance of intelligent life existing elsewhere in the Universe or of you winning the lottery?”

 

 

 

Ask, "what variables CAN we manipulate in the Drake Equation so that there is a better chance of YOU winning the lottery?" or, "what variables MUST we manipulate in the Drake Equation so that there is a better chance of YOU winning the lottery?".

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Activity 1/ Part A

Simulate the experimental probability of a lottery game (such as Pick 3) where you, the participants, model the game with decahedron dice in cooperative groups of 3.

 

 

 

Exploration

  1. Each group of three must have 3 decahedron dice**, and must determine who is the recorder, dice roller, and digit chooser.
  2. The digit chooser chooses a three digit number. Write your number below.

Note: we are not 'boxing' the number, each number rolled must be kept in order.

  1. Dice roller, roll the 3 dice (one at a time) 20 times. Recorder, keep track of the data acquired in an organized fashion.
  2. What was the experimental probability of your group winning with the digit chooser's number? Show your work below.
  3. Now determine the theoretical probability of your group winning with the digit chooser's number. Write it below.
  4. Explain how your answers to #4 and #5 are the same or different.

  5. How could we change or modify our experiment so that the experimental probability would be closer to the theoretical probability?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Activity 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Activity 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Evaluation

Have students write a journal entry focusing on the chance of intelligent life existing elsewhere in the Universe.

 

 

 

 

 

 

 

 

 

References:  Original lesson plan written by Kara C. Granger, Maria Carillo High School, Santa Rosa, CA (available at http://imagine.gsfc.nasa.gov/docs/teachers/lessons/lotto/lotto.html).  It has been edited, revised for content, and adapted to “5-E” format by Ellen Lukasik, UTeach Program, University of Texas at Austin.

 

 

 

 

 

 

 

 

 

 

Lotto or Life: What are the Chances?

Activity 1/ Part A

 

Now, before we go further with the discussion of life existing elsewhere in the Universe,

we will establish some understandings about the lottery so that we can compare the

likelihood of intelligent life existing elsewhere in the Universe and winning the lottery at

the end of our lesson.

 

Let's simulate the experimental probability of a lottery game (such as Pick 3) where you,

the participants, model the game with decahedron dice in cooperative groups of 3.

 

 

Procedure

1. Each group of three must have 3 decahedron dice, and must determine who is the

recorder, dice roller, and digit chooser.

 

 

 

2. The digit chooser chooses a three digit number. Write your number below.

Note: we are not 'boxing' the number, each number rolled must be kept in order.

 

 

 

3. Dice roller, roll the 3 dice (one at a time) 20 times. Recorder, keep track of the data

acquired in an organized fashion.

 

 

 

 

 

4. What was the experimental probability of your group winning with the digit chooser's

number? Show your work below.

 

 

 

 

 

 

5. Now determine the theoretical probability of your group winning with the digit

chooser's number. Write it below.

 

 

 

 

 

 

6. Explain how your answers to #4 and #5 are the same or different.

 

 

 

 

 

 

7. How could we change or modify our experiment so that the experimental probability

would be closer to the theoretical probability?

 

Lotto or Life: What are the Chances?

Activity 1/ Part B

 

Next, we look at a lottery game which is slightly more complicated than

Pick 3. This time we will examine the theoretical probability of winning a

multiple digit game such as "Lotto". If we chose a certain series of 7

numbers, each less than 40 as is allowed in "Lotto", what would be our

chances of winning?

 

 

If we calculate,

W=a * b * c * d * e * f * g

where,

 

W = The probability of winning the "Lotto"

 

a = The probability of 'getting' the first number

 

b = The probability of 'getting' the second number

 

c = The probability of 'getting' the third number

 

d = The probability of 'getting' the fourth number

 

e = The probability of 'getting' the fifth number

 

f = The probability of 'getting' the sixth number

 

g = The probability of 'getting' the seventh number

 

If we know that each number is one of forty and a given number cannot be

called twice, we substitute

 

W=a * b * c * d * e * f * g

 

W= (1/40) (1/39) (1/38) (1/37) (1/36) (1/35) (1/34)

 

W= 1/93,963,542,400

 

an extremely small number!

 

 

 

 

Lotto or Life: What are the Chances ? 

Activity 3

Is there a way to estimate the number of technologically advanced

civilizations that might exist in our Galaxy? While working at the National

Radio Astronomy Observatory in Green Bank, West Virginia, Dr. Frank

Drake conceived a means to mathematically estimate the number of worlds

that might harbor beings with technology that could communicate across the

vastness of interstellar space. The Drake Equation, as it came to be known,

was formulated in 1961 and is generally accepted by the scientific

community.

 

N = fs * fp * ne * fld * fi * fc * fl

where,

N = The number of communicative civilizations in the Milky Way

 

fs = The number of stars in the Milky Way

 

fp = The fraction of those stars with planets (Current evidence

indicates that planetary systems may be common for stars like the

Sun.)

 

ne = The number of Earth-like worlds per planetary system

 

fld = The fraction of those Earth-like planets where life actually

develops

 

fi = The fraction of life sites where intelligence develops

 

fc = The fraction of communicative planets (those on which

electromagnetic communications technology

develops)

 

fl = The fraction of a planet's lifetime that has a technological

civilization

 

If we substitute,

N = fs * fp * ne * fld * fi * fc * fl

 

N = 400 billion (1/4) 2 (1/2) (1/10) (1/10) (1/100 million)

N ~ 10 technological civilizations in just the Milky Way Galaxy!

 

What would we do to determine the number of technological civilizations

that exists in the Universe? Multiply this by billions; the number of galaxies

in the Universe!