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Introduction

In probability theory event are either independent or dependent. This chapter describes the differences and how each type of event is worked with.

Definitions

Two events are independent if knowing something about the value of one event does not give any information about the value of the second event. For example, the event of getting a "1" when a die is rolled and the event of getting a "1" the second time it is thrown are independent.

Independent Events

Two events A and B are independent if when one of them happens, it doesn't affect whether the one happens or not.

The probability of two independent events occurring, P ( A B ) , is given by:

P ( A B ) = P ( A ) × P ( B )

What is the probability of rolling a 1 and then rolling a 6 on a fair die?

  1. Event A is rolling a 1 and event B is rolling a 6. Since the outcome of the first event does not affect the outcome of the second event, the events are independent.

  2. The probability of rolling a 1 is 1 6 and the probability of rolling a 6 is 1 6 .

    Therefore, P ( A ) = 1 6 and P ( B ) = 1 6 .

  3. P ( A B ) = P ( A ) × P ( B ) = 1 6 × 1 6 = 1 36

    The probability of rolling a 1 and then rolling a 6 on a fair die is 1 36 .

Consequently, two events are dependent if the outcome of the first event affects the outcome of the second event.

A cloth bag has four coins, one R1 coin, two R2 coins and one R5 coin. What is the probability of first selecting a R1 coin followed by selecting a R2 coin?

  1. Event A is selecting a R1 coin and event B is next selecting a R2. Since the outcome of the first event affects the outcome of the second event (because there are less coins to choose from after the first coin has been selected), the events are dependent.

  2. The probability of first selecting a R1 coin is 1 4 and the probability of next selecting a R2 coin is 2 3 (because after the R1 coin has been selected, there are only three coins to choose from).

    Therefore, P ( A ) = 1 4 and P ( B ) = 2 3 .

  3. The same equation as for independent events are used, but the probabilities are calculated differently.

    P ( A B ) = P ( A ) × P ( B ) = 1 4 × 2 3 = 2 12 = 1 6

    The probability of first selecting a R1 coin followed by selecting a R2 coin is 1 6 .

Identification of independent and dependent events

Use of a contingency table

A two-way contingency table (studied in an earlier grade) can be used to determine whether events are independent or dependent.

two-way contingency table

A two-way contingency table is used to represent possible outcomes when two events are combined in a statistical analysis.

For example we can draw and analyse a two-way contingency table to solve the following problem.

A medical trial into the effectiveness of a new medication was carried out. 120 males and 90 females responded. Out of these 50 males and 40 females responded positively to the medication.

  1. Was the medication's success independent of gender? Explain.
  2. Give a table for the independence of gender results.
  1. Male Female Totals
    Positive result 50 40 90
    No Positive result 70 50 120
    Totals 120 90 210
  2. P(male).P(positive result)= 120 210 = 0 , 57

    P(female).P(positive result)= 90 210 = 0 , 43

    P(male and positive result)= 50 210 = 0 , 24

  3. P(male and positive result) is the observed probability and P(male).P(positive result) is the expected probability. These two are quite different. So there is no evidence that the medication's success is independent of gender.

  4. To get gender independence we need the positive results in the same ratio as the gender. The gender ratio is: 120:90, or 4:3, so the number in the male and positive column would have to be 4 7 of the total number of patients responding positively which gives 51,4. This leads to the following table:

    Male Female Totals
    Positive result 51,4 38,6 90
    No Positive result 68,6 51,4 120
    Totals 120 90 210

Questions & Answers

what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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
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?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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preparation of nanomaterial
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Other chapter Q/A we can ask
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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