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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)
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**With contributions from Roberta Bloom

Questions & Answers

how can chip be made from sand
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no can't
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Maira Reply
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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RAW Reply
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yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
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what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Smarajit Reply
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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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