# 4.13 Probability topics: probability lab

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This module presents students with a lab exercise allowing them to apply their understanding of Probability. In an experiment using M&Ms candies, students will calculate and compare the theoretical and empirical probabilities of drawing particular color candies at random, with and without replacement.

Class time:

Names:

## Student learning outcomes:

• The student will use theoretical and empirical methods to estimate probabilities.
• The student will appraise the differences between the two estimates.
• The student will demonstrate an understanding of long-term relative frequencies.

## Do the experiment:

Count out 40 mixed-color M&M’s® which is approximately 1 small bag’s worth (distance learning classes using the virtual lab would want to count out 25 M&M’s®). Record the number of each color in the "Population" table. Use the information from this table to complete the theoretical probability questions. Next, put the M&M’s in a cup. The experiment is to pick 2 M&M’s, one at a time. Do not look at them as you pick them. The first time through, replace the first M&M before picking the second one. Record the results in the “With Replacement” column of the empirical table. Do this 24 times. The second time through, after picking the first M&M, do not replace it before picking the second one. Then, pick the second one. Record the results in the “Without Replacement” column section of the "Empirical Results" table. After you record the pick, put both M&M’s back. Do this a total of 24 times, also. Use the data from the "Empirical Results" table to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do not multiply out any fractions.

Population
Color Quantity
Yellow (Y)
Green (G)
Blue (BL)
Brown (B)
Orange (O)
Red (R)
Theoretical probabilities
With Replacement Without Replacement
$P\left(\text{2 reds}\right)$
$P\left({R}_{1}{B}_{2}\text{OR}{B}_{1}{R}_{2}\right)$
$P\left({R}_{1}\text{AND}{G}_{2}\right)$
$P\left({G}_{2}\text{|}{R}_{1}\right)$
$P\left(\text{no yellows}\right)$
$P\left(\text{doubles}\right)$
$P\left(\text{no doubles}\right)$
Empirical results
With Replacement Without Replacement
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
Empirical probabilities
With Replacement Without Replacement
$P\left(\text{2 reds}\right)$
$P\left({G}_{2}\text{|}{R}_{1}\right)$
$P\left(\text{no yellows}\right)$
$P\left(\text{doubles}\right)$
$P\left(\text{no doubles}\right)$

## Discussion questions

1. Why are the “With Replacement” and “Without Replacement” probabilities different?
2. Convert $\text{P(no yellows)}$ to decimal format for both Theoretical “With Replacement” and for Empirical “With Replacement”. Round to 4 decimal places.
• Theoretical “With Replacement”: $\text{P(no yellows)}=$
• Empirical “With Replacement”: $\text{P(no yellows)}=$
• Are the decimal values “close”? Did you expect them to be closer together or farther apart? Why?
3. If you increased the number of times you picked 2 M&M’s to 240 times, why would empirical probability values change?
4. Would this change (see (3) above) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?
5. Explain the differences in what $P\left({G}_{1}\text{AND}\phantom{\rule{2pt}{0ex}}{R}_{2}\right)$ and $P\left({R}_{1}\text{|}{G}_{2}\right)$ represent. Hint: Think about the sample space for each probability.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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