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This important characteristic of probability experiments is the known as the Law of Large Numbers : as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes don't happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.) The Law of Large Numbers will be discussed again in Chapter 7.

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair , or biased . Two math professors in Europe had their statistics students test the Belgian 1 Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos have a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later in this chapter we will learn techniques to use to work with probabilities for events that are not equally likely.

"or" event:

An outcome is in the event A   OR   B if the outcome is in A or is in B or is in both A and B . For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8} . A   OR   =  {1, 2, 3, 4, 5, 6, 7, 8} . Notice that 4 and 5 are NOT listed twice.

"and" event:

An outcome is in the event A AND B if the outcome is in both A and B at the same time.For example, let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8} , respectively. Then A AND B = { 4 , 5 } .

The complement of event A is denoted A' (read "A prime"). A' consists of all outcomes that are NOT in A . Notice that P(A) + P(A') = 1 . For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4} . Then, A' = {5, 6}. P(A) = 4 6 , P(A') = 2 6 , and P(A) + P(A') = 4 6 + 2 6 = 1

The conditional probability of A given B is written P(A|B) . P(A|B) is the probability that event A will occur given that the event B has already occurred. A conditional reduces the sample space . We calculate the probability of A from the reduced sample space B . The formula to calculate P(A|B) is

P(A|B)= P(A AND B) P(B)

where P(B) is greater than 0.

For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6} . Let A = face is 2 or 3 and B = face is even (2, 4, 6). To calculate P(A|B) , we count the number of outcomes 2 or 3 in the sample space B = {2, 4, 6} . Then we divide that by the number of outcomes in B (and not S ).

We get the same result by using the formula. Remember that S has 6 outcomes.

P(A|B) = P(A and B) P(B) = (the number of outcomes that are 2 or 3 and even in S) / 6 (the number of outcomes that are even in S) / 6 = 1/6 3/6 = 1 3

Understanding terminology and symbols

It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.

In a particular college class, there are male and female students. Some students have long hair and some students have short hair.Write the symbols for the probabilities of the events for parts (a) through (j) below. (Note that you can't find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)

  • Let F be the event that a student is female.
  • Let M be the event that a student is male.
  • Let S be the event that a student has short hair.
  • Let L be the event that a student has long hair.
  • The probability that a student does not have long hair.
  • The probability that a student is male or has short hair.
  • The probability that a student is a female and has long hair.
  • The probability that a student is male, given that the student has long hair.
  • The probability that a student has long hair, given that the student is male.
  • Of all the female students, the probability that a student has short hair.
  • Of all students with long hair, the probability that a student is female.
  • The probability that a student is female or has long hair.
  • The probability that a randomly selected student is a male student with short hair.
  • The probability that a student is female.
  • P(L')=P(S)
  • P(M or S)
  • P(F and L)
  • P(M|L)
  • P(L|M)
  • P(S|F)
  • P(F|L)
  • P(F or L)
  • P(M and S)
  • P(F)
Got questions? Get instant answers now!

**With contributions from Roberta Bloom

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
Nerisha Reply
Practice Key Terms 7

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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