Lotto or Life: What Are the Chances?
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.
- Program #12 in the Carl Sagan "Cosmos" series, Imagine
the Universe! web site or CD-Rom.
- 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
- Students should have a basic understanding of probability
and space science.
- For more information about space science, see the
following web sites:
- For additional information on probability, the best source
would be any middle school mathematics text (or teacher's
"Are we alone?"
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, 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
For more information and details on the formation of
planets and life, see the "Timeline of the Universe" at
For further insights on the Structure and Evolution of the
Universe, see "Imagine the Universe!" at
(pdf version available)
NOTE: Reading pdf files requires the Adobe Acrobat Reader, which is available for free download from http://www.adobe.com/products/acrobat/readstep.html.
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.
- Each group of three must have 3 decahedron dice**,
and must determine who is the recorder, dice roller,
and digit chooser.
- 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.
- Dice roller, roll the 3 dice (one at a time) 20 times.
Recorder, keep track of the data acquired in an
- What was the experimental probability of your group
winning with the digit chooser's number? Show your
- Now determine the theoretical probability of your
group winning with the digit chooser's number. Write
- Explain how your answers to #4 and #5 are the same
- How could we change or modify our experiment so
that the experimental probability would be closer to
the theoretical probability?
(pdf version available)
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
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)
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.
Focus for Viewing
Say: Now that we have determined both experimental and
theoretical probability of winning the lottery, we will begin
our theoretical probability discussion about the chances of
intelligent life existing elsewhere in the Universe.
Viewing Activity: Program #12 in the Carl Sagan "Cosmos"
- START the video 90 minutes into the tape at the point
where you see and hear the Arecibo Observatory being discussed.
- PAUSE when you see Carl Sagan begin to explain the
- FOCUS: Ask students to pay close attention to each
value substituted in the Drake Equation. RESUME.
- PAUSE when Carl Sagan has solved the equation and
obtained an answer.
- 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.
- STOP when Carl Sagan says "...enormously older and
wiser than we"."
(pdf version available)
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
N = fs * fp * ne * fld * fi * fc * fl
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
ne = The number of Earth-like worlds per planetary
fld = The fraction of those Earth-like planets where life
fi = The fraction of life sites where intelligence
fc = The fraction of communicative planets (those on which
electromagnetic communications technology develops)
fl = The fraction of a planet's lifetime that has a
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
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!
Using transparencies of the two calculations (lottery and
the Drake Equation) we are prepared to answer the
following question; "Is there a better chance of intelligent
life existing elsewhere in the Universe or of you winning
Have students write a journal entry focusing on the chance
of intelligent life existing elsewhere in the Universe. 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?".
Other assessments could include creating other lottery
games where players are 'guaranteed' to win (modeling the
likelihood of intelligent life existing elsewhere in the
- Research the number of Sun-like solar systems
discovered thus far.
- Invite a lottery official to your classroom to discuss
the state/multi-state lottery games.
- Have the science teachers in your school discuss the
probability of inheriting certain characteristics (as
modeled by the Punnett Square).
- 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