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In this article, we will solve some examples related to probability. So, let us get started.

## Example 1

In a class of 32 students, 25 take mathematics, 22 take science, and 15 students take both mathematics and science. Find the probability that a randomly selected student:

a) takes mathematics only

b) takes science only

c) takes mathematics

d) takes science

e) takes mathematics and science

f) takes mathematics given that he already takes science

### Solution

This question has many parts. We will solve each part one by one, but first, we will draw a Venn diagram from the information in the question like this:

### Part a

Number of students who take mathematics only = 10

Total number of students in the class = 32

Probability that a randomly selected student takes mathematics only = P(A) =

### Part b

Number of students who take science only = 7

Total number of students in the class = 32

Probability that a randomly selected student takes science only = P(B) =

### Part c

Number of students who take mathematics  = 10 + 15 = 25

Total number of students in the class = 32

Probability that a randomly selected student takes mathematics = P(C) =

### Part d

Number of students who take science  = 7 + 15 = 22

Total number of students in the class = 32

Probability that a randomly selected student takes science = P(D) =

### Part e

Number of students who take both mathematics and science = 15

Total number of students in the class = 32

The probability that randomly selected student takes both mathematics and science =

### Part f

Number of students who take science = 7 + 15 = 22

Number of students who take mathematics and science = =  15

Probability that a randomly selected student takes mathematics, given that he already takes science =

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£48
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1st lesson free!
4.9 (29 reviews)
Paolo
£30
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1st lesson free!
5 (16 reviews)
Jamie
£25
/h
1st lesson free!
5 (16 reviews)
Harinder
£15
/h
1st lesson free!
5 (17 reviews)
Matthew
£30
/h
1st lesson free!
4.9 (12 reviews)
Petar
£40
/h
1st lesson free!
4.9 (31 reviews)
Sehaj
£25
/h
1st lesson free!
5 (24 reviews)
Shane
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1st lesson free!

## Example 2

Two friends go hunting. The first friend kills an average of 3 animals every 6 shots and the friend kills two animals every 7 shots. If the two fired at the same animal at the same time, find the probability that they will obtain a kill.

### Solution

Suppose A be an event that the first friend obtains a kill. Then the probability of obtaining a kill of the first friend will be represented by P(A).

Probability of the first friend obtaining a kill = P(A) =

Suppose B be an event that the second friend obtains a kill. Then the probability of obtaining a kill of the second friend will be represented by P(B).

Probability of the second friend obtaining a kill = P(B) =

The probability of first and second friend obtaining a kill = =

Probability that both the friends obtain a kill = =

=

## Example 3

The probability that the annual event will be held on a specific day is 0.57. Find the probability that:

a) the event will not be held on that day

b) the event will take place or not take place on that day

### Part a

The probability of event taking place at a certain day = P(E) = 0.57

The probability of event not taking place at a certain day = P(E)' = 1 - P(E) = 1 - 0.57 = 0.43

### Part b

Probability of event taking place at a certain day = P(E) = 0.57

Probability of event not taking place at a certain day = P(E)' = 1 - P(E) = 1 - 0.57 = 0.43

The probability of event taking place or not taking place on a certain day = P(E U E') = 0.57 + 0.43 = 1.00

## Example 4

A class is divided into three groups on the basis of the preferred subjects combinations of the students. The first group has 20 students, out of which 5 like mathematics. The second group has 12 students, out of which 7 like mathematics. The third group has 18 students, out of which 3 like mathematics. Find the probability of selecting a student randomly from one of the three groups who like mathematics.

### Solution

Total number of groups = 3

The number of groups from which a student who likes mathematics will be selected = 1

Probability = P (B_i) =

Total number of students in the first group = 20

Number of students who like mathematics = 5

Probability of choosing a student who likes mathematics from the first group =

Total number of students in the second group = 12

Number of students in the second group who like mathematics  = 7

Probability of choosing a student who likes mathematics from the second group =

Total number of students in the third group = 18

Number of students who like mathematics in the third group = 3

Probability of choosing a student who like mathematics from the third group =

The probability of choosing the student who likes mathematics from one of the three groups will be calculated using the total probability rule.

P (girl) =

## Example 5

In a group of 50 people, 35 own a car, 25 own a bike, and 20 own both a car and a bike. 10 people neither own a car nor a bike. Find the probability that a randomly selected person:

a) owns a car only

b) owns a bike only

c) owns a car

d) owns a bike

e) owns a car and a bike

f) neither owns a car nor a bike

g) owns a car given that he already has a bike

### Solution

This question has many parts. We will solve each part one by one, but first, we will draw a Venn diagram from the information in the question like this:

### Part a

Number of people who have a car only = 15

Total number of people = 50

Probability that a randomly selected person owns a car only = P(A) =

### Part b

Number of people who have bike only = 5

Total number of people = 50

Probability that a randomly selected person owns a bike only = P(B) =

### Part c

Number of people who own a car  = 20 + 15 = 35

Total number of people = 50

Probability that a randomly selected person owns a car = P(C) =

### Part d

Number of people who own bike  = 5 + 20 = 25

Total number of people = 50

Probability that a randomly selected person owns a bike = P(D) =

### Part e

Number of students who take own a car and a bike both = 20

Total number of people = 50

The probability that a randomly selected person owns a car and a bike =

### Part f

Number of people who neither own a car nor a bike = 10

Total number of people = 50

The probability that a randomly selected person neither owns a car nor a bike =

### Part g

Probability that a randomly selected person owns a car given that he already has a bike = P(A|B) =

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