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Probability models

  1. A bag contains 6 red, 3 blue, 2 green and 1 white balls. A ball is picked at random. What is the probablity that it is:
    1. red
    2. blue or white
    3. not green (hint: think 'complement')
    4. not green or red?
  2. A card is selected randomly from a pack of 52. What is the probability that it is:
    1. the 2 of hearts
    2. a red card
    3. a picture card
    4. an ace
    5. a number less than 4?
  3. Even numbers from 2 -100 are written on cards. What is the probability of selecting a multiple of 5, if a card is drawn at random?

Probability identities

The following results apply to probabilities, for the sample space S and two events A and B , within S .

P ( S ) = 1
P ( A B ) = P ( A ) × P ( B )
P ( A B ) = P ( A ) + P ( B ) - P ( A B )

We can demonstrate this last result using a Venn diagram. The union of A and B is the set of all elements in A or in B or in both.

The probability of event A occurring is given by P(A) and the probability of event B occurring is given by P(B) . However, if we look closely at the circle representing either of these events, we notice that the probability includes a small part of the other event. So event A includes a bit of event B and vice versa. This is shown in the following figure:

And then we observe that this small bit is simply the intersection of the two events.

So to find the probability of P ( A B ) we notice the following:

  • We can add P ( A ) and P ( B )
  • Doing this counts the intersection twice, once in P ( A ) and once in P ( B ) .
So if we simply subtract the probability of the intersection, then we will find the total probability of the union: P ( A B ) = P ( A ) + P ( B ) - P ( A B )

What is the probability of selecting a black or red card from a pack of 52 cards

  1. P ( S ) = n ( E ) n ( S ) = 52 52 = 1 because all cards are black or red!

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What is the probability of drawing a club or an ace with one single pick from a pack of 52 cards

  1. P ( club ace ) = P ( club ) + P ( ace ) - P ( club ace )
  2. = 1 4 + 1 13 - 1 4 × 1 13 = 1 4 + 1 13 - 1 52 = 16 52 = 4 13

    Notice how we have used P ( C A ) = P ( C ) + P ( A ) - P ( C A ) .

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The following video provides a brief summary of some of the work covered so far.

Khan academy video on probability

Probability identities

Answer the following questions

  1. Rory is target shooting. His probability of hitting the target is 0,7 . He fires five shots. What is the probability that all five shots miss the center?
  2. An archer is shooting arrows at a bullseye. The probability that an arrow hits the bullseye is 0,4 . If she fires three arrows, what is the probability that all the arrows hit the bullseye?
  3. A dice with the numbers 1,3,5,7,9,11 on it is rolled. Also a fair coin is tossed. What is the probability that:
    1. A tail is tossed and a 9 rolled?
    2. A head is tossed and a 3 rolled?
  4. Four children take a test. The probability of each one passing is as follows. Sarah: 0,8 , Kosma: 0,5 , Heather: 0,6 , Wendy: 0,9 . What is the probability that:
    1. all four pass?
    2. all four fail?
  5. With a single pick from a pack of 52 cards what is the probability that the card will be an ace or a black card?

Mutually exclusive events

Two events are called mutually exclusive if they cannot be true at the same time.

Examples of mutually exclusive events are:

  1. A die landing on an even number or landing on an odd number.
  2. A student passing or failing an exam
  3. A tossed coin landing on heads or landing on tails

This means that if we examine the elements of the sets that make up A and B there will be no elements in common. Therefore, A B = (where refers to the empty set). Since, P ( A B ) = 0 , equation [link] becomes:

P ( A B ) = P ( A ) + P ( B )

for mutually exclusive events.

We can represent mutually exclusive events on a Venn diagram. In this case, the two circles do not touch each other, but are instead completely separate parts of the sample space.

Venn diagram for mutually exclusive events

Mutually exclusive events

  1. A box contains coloured blocks. The number of each colour is given in the following table.
    Colour Purple Orange White Pink
    Number of blocks 24 32 41 19
    A block is selected randomly. What is the probability that the block will be:
    1. purple
    2. purple or white
    3. pink and orange
    4. not orange?
  2. A small private school has a class with children of various ages. The table gies the number of pupils of each age in the class.
    3 years female 3 years male 4 years female 4 years male 5 years female 5 years male
    6 2 5 7 4 6
    If a pupil is selceted at random what is the probability that the pupil will be:
    1. a female
    2. a 4 year old male
    3. aged 3 or 4
    4. aged 3 and 4
    5. not 5
    6. either 3 or female?
  3. Fiona has 85 labeled discs, which are numbered from 1 to 85. If a disc is selected at random what is the probability that the discnumber:
    1. ends with 5
    2. can be multiplied by 3
    3. can be multiplied by 6
    4. is number 65
    5. is not a multiple of 5
    6. is a multiple of 4 or 3
    7. is a multiple of 2 and 6
    8. is number 1?

Random experiments

  1. Let S denote the set of whole numbers from 1 to 16, X denote the set of even numbers from 1 to 16 and Y denote the set of prime numbers from 1 to 16
    1. Draw a Venn diagram accurately depicting S , X and Y .
    2. Find n ( S ) , n ( X ) , n ( Y ) , n ( X Y ) , n ( X Y ) .
  2. There are 79 Grade 10 learners at school. All of these take either Maths, Geography or History. The number who take Geography is41, those who take History is 36, and 30 take Maths. The number who take Maths and History is 16; the number who take Geography and History is 6, and there are8 who take Maths only and 16 who take only History.
    1. Draw a Venn diagram to illustrate all this information.
    2. How many learners take Maths and Geography but not History?
    3. How many learners take Geography only?
    4. How many learners take all three subjects?
  3. Pieces of paper labelled with the numbers 1 to 12 are placed in a box and the box is shaken. One piece of paper is taken out andthen replaced.
    1. What is the sample space, S ?
    2. Write down the set A , representing the event of taking a piece of paper labelled with a factor of 12.
    3. Write down the set B , representing the event of taking a piece of paper labelled with a prime number.
    4. Represent A , B and S by means of a Venn diagram.
    5. Find
      1. n ( S )
      2. n ( A )
      3. n ( B )
      4. n ( A B )
      5. n ( A B )
    6. Is n ( A B ) = n ( A ) + n ( B ) - n ( A B ) ?

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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