<< Chapter < Page Chapter >> Page >


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


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

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Other chapter Q/A we can ask
Moahammedashifali Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 11 maths' conversation and receive update notifications?