# 0.11 Independent and dependent events

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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, $P\left(A\cap B\right)$ , is given by:

$P\left(A\cap B\right)=P\left(A\right)×P\left(B\right)$

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 $\frac{1}{6}$ and the probability of rolling a 6 is $\frac{1}{6}$ .

Therefore, $P\left(A\right)=\frac{1}{6}$ and $P\left(B\right)=\frac{1}{6}$ .

3. $\begin{array}{ccc}\hfill P\left(A\cap B\right)& =& P\left(A\right)×P\left(B\right)\hfill \\ & =& \frac{1}{6}×\frac{1}{6}\hfill \\ & =& \frac{1}{36}\hfill \end{array}$

The probability of rolling a 1 and then rolling a 6 on a fair die is $\frac{1}{36}$ .

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

Therefore, $P\left(A\right)=\frac{1}{4}$ and $P\left(B\right)=\frac{2}{3}$ .

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

$\begin{array}{ccc}\hfill P\left(A\cap B\right)& =& P\left(A\right)×P\left(B\right)\hfill \\ & =& \frac{1}{4}×\frac{2}{3}\hfill \\ & =& \frac{2}{12}\hfill \\ & =& \frac{1}{6}\hfill \end{array}$

The probability of first selecting a R1 coin followed by selecting a R2 coin is $\frac{1}{6}$ .

## 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)= $\frac{120}{210}=0,57$

P(female).P(positive result)= $\frac{90}{210}=0,43$

P(male and positive result)= $\frac{50}{210}=0,24$

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 $\frac{4}{7}$ 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

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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