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
$\mathrm{P(A)}$ and the probability of event B occurring is given by
$\mathrm{P(B)}$ . 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
$P(A\cup B)$ we notice the following:
We can add
$P\left(A\right)$ and
$P\left(B\right)$
Doing this counts the intersection twice, once in
$P\left(A\right)$ and once in
$P\left(B\right)$ .
So if we simply subtract the probability of the intersection, then we will find the total probability of the union:
$$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$$
What is the probability of selecting a black or red
card from a pack of 52 cards
$P\left(S\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{52}{52}=1$ because all cards are black or
red!
The following video provides a brief summary of some of the
work covered so far.
Probability identities
Answer the following questions
Rory is target shooting. His probability of hitting the target is
$\mathrm{0,7}$ . He fires five shots. What is the probability
that all five shots miss the center?
An archer is shooting arrows at a bullseye. The
probability that an arrow hits the bullseye is
$\mathrm{0,4}$ . If she fires three arrows, what is the
probability that all the arrows hit the bullseye?
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:
A tail is tossed and a 9 rolled?
A head is tossed and a 3 rolled?
Four children take a test. The probability of each one
passing is as follows. Sarah:
$\mathrm{0,8}$ , Kosma:
$\mathrm{0,5}$ , Heather:
$\mathrm{0,6}$ ,
Wendy:
$\mathrm{0,9}$ . What is the probability that:
all four pass?
all four fail?
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:
A die landing on an even number or landing on an odd
number.
A student passing or failing an exam
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\cap B=\varnothing $ (where
$\varnothing $ refers to the empty set). Since,
$P(A\cap B)=0$ ,
equation
[link] becomes:
$$P(A\cup B)=P\left(A\right)+P\left(B\right)$$
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.
Mutually exclusive events
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:
purple
purple or white
pink and orange
not orange?
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:
a female
a 4 year old male
aged 3 or 4
aged 3 and 4
not 5
either 3 or female?
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:
ends with 5
can be multiplied by 3
can be multiplied by 6
is number 65
is not a multiple of 5
is a multiple of 4 or 3
is a multiple of 2 and 6
is number 1?
Random experiments
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
Draw a Venn
diagram accurately depicting
$S$ ,
$X$ and
$Y$ .
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.
Draw a Venn diagram to illustrate
all this information.
How many learners take Maths and Geography but not
History?
How many learners take Geography only?
How many learners take all three subjects?
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.
What is the sample space,
$S$ ?
Write down the set
$A$ , representing the event of taking a
piece of paper labelled with a factor of 12.
Write down the set
$B$ , representing the event of taking a
piece of paper labelled with a prime number.
Represent
$A$ ,
$B$ and
$S$ by means of a Venn diagram.
Find
$n\left(S\right)$
$n\left(A\right)$
$n\left(B\right)$
$n(A\cap B)$
$n(A\cup B)$
Is
$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$ ?
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what is differents between GO and RGO?
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analytical skills graphene is prepared to kill any type viruses .
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The nanotechnology is as new science, to scale nanometric
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Damian
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