The types of probability are as follows:

● Theoretical probability: The possibility of something occurring is regarded as theoretical probability. It is completely based on reasoning.

 ● Experimental probability: Experimental probability is also known as observatory probability. It is calculated by comparing the possible outcomes to the total number of trials.

● Axiomatic probability: Axiomatic probability has a set of rules that helps in estimating the occurrence/non-occurrence of any event.

To understand probability better, it is important to get accustomed to the related terms:

Events

It is one of the most common terms used in the theories of probability. Events are occurrences in which there are different types of expected outcomes. The different types of events are:

 

> Exhaustive events

> Independent events

> Dependent events

> Impossible events

> Sure events

> Mutually exclusive events

Experiment

 

In probability, an experiment is regarded as an activity whose outcome is still unknown. Experiments can have favourable or unfavourable outcomes.

Random experiments are those in which there are many possible outcomes and when the outcome of one experiment cannot be predicted.

If you conduct an experiment with a set number of repetitions, the individual experiments are called trials. 

Outcome

The result of a trial or experiment is regarded as the event’s outcome. The outcome here is also divided into types:

> Possible outcome: All the potential outcomes of an experiment or trial are regarded as possible outcomes.

> Equally likely outcomes: Outcomes with an equal possibility of occurrence are regarded as equally likely outcomes.

Sample space

The result of all the trials of an experiment is called the sample space. It is a set of all possible outcomes.

These are the essential terms related to or used in probability.

Some of the critical questions and their NCERT Solutions for Class 9 Maths – Chapter 15

are as follows:

Question 1

A container contains 50 berries and 150 nuts. Later it was discovered that half of the berries and nuts were rotten. If one of the items is chosen at random, what is the probability of it being rotten?

Question 2

The probability of predicting the correct answer to a question is x/2. If the probability of being unable to guess the correct answer is 2/3, find out x.

Question 3

A big bag has a certain number of balls in it. There are:

● X white balls

● Y red balls

● Z blue balls

When a ball is drawn at random, what is the probability that the chosen ball is blue?

Question 4

An athlete’s probability of winning a race is almost 1/6 less than twice the probability of losing the race. Calculate the probability of the athlete winning the race.

Question 5

Calculate the probability of an event’s occurrence if the probability of that event not occurring is 0.56.

Solutions

The solutions to the above questions are as follows:

Solution 1

The total number of nuts and berries present in the container: 50 + 150 = 200

The rotten nuts and berries among them: 1/2 × 200 = 100

The probability of the chosen product is actually rotten: 100/200 = 1/2

Answer: 1/2

Solution 2

We know that the probability of guessing the correct answer is x/2

We also know that the probability of not guessing the correct answer is 2/3

Therefore, according to the problem, x is:

(x/2) + (2/3) = 1

3x + 4 = 6

3x = 2

Therefore, x = 2/3

Answer: 2/3

Solution 3

We know that the total number of blue balls is Z

Total number of ways: X + Y + Z

Now, the total number of blue balls is

Z/( X + Y + Z )

Answer: Z/( X + Y + Z )

Solution 4

Let us consider the probability of winning the race to be p

Then, the probability of losing the race will be: 1 - p

Now, according to the problem:

p = 2 (1 - p ) - 1/6

6p = 12 -12p - 1

18p = 11

p = 11/18

Answer: The probability of winning the race is 11/18

Solution 5

Let the probability of the event not occurring be P (not E) = 0. 56

Let the probability of the occurrence of the event be P (E)

Now, according to the problem,

P (not E) + P (E) = 1

P (E) = 1 - P ( not E)

P (E) = 1 - 0.56

P (E) = 0.44

Therefore, the probability of the event’s occurrence is 0.44.

Question 1

Among the 35 students of a class participating in a debate competition, ten are girls. What is the probability that the winner of the competition is a boy?

a) 1

b) 27

c) 37

d) 57

Answer: d) 57

Question 2

A total of five balls are present in a bag, in the following colours: white, red, violet, brown, and yellow. If one ball is drawn from the bag, calculate the probability that the removed ball is brown.

a) 45

 b) 14

 c) 15

 d) 120

Answer: c) 15

Question 3

If P(e) or the probability of an event’s occurrence is calculated to be 0.25, what is the value of P(not e) or non-occurrence of an event?

a) 0.5

b) 1

c) 0

d) 0.75

Answer: d) 0.75

Question 4

What is the sum of the probabilities of all the events in a trial?

a) Less than one

b) More than one

c) Lying between zero and one

d) Directly one

Answer: d)

Question 5

A class has 15 boys and ten girls. When three students are selected at random for a competition, calculate the probability of one girl and two boys being selected.

a) 1/40

b) 1/2

c) 21/46

d) 7/41

Answer: c) 21/46

Question 6

Two friends, Rekha and Suresh, appeared for an exam. Let’s consider A the event of Rekha’s selection, whereas B is the event of Suresh’s selection. Hence, the probability of A is 2/5, while the probability of B is 3/7. Therefore, find the probability that both Rekha and Suresh are selected.

a) 35/36

b) 5/35

c) 5/12

d) 6/35

Answer: d) 6/35

Question 7

Miss Piu speaks the truth in almost 2/5 cases, whereas Mr. Ram lies in almost 3/7 cases. Now calculate the percentage of cases where Miss Piu and Mr. Ram contradict each other in accepting a fact.

a) 72.6%

b) 51.4%

c) 62.3%

d) 47.5%

Answer: b) 51.4%

Question: 8

The names of students from different sections of the class are being taken: five students from section A, six students from section B, and seven students from section C. The ages of the 18 selected students are different. Now, one name is being selected from it, and it was seen that it was a student from section B. Calculate the probability that it was the name of the youngest student from section B.

a) 1/18

b) 1/15

c) 1/6

d) 1/12

Answer: c) 1/6

Some of the uses of probability from Class 9 Maths Chapter 15 are as follows:

• Helps in predicting the weather

Meteorologists use the science of probability to predict weather conditions and upcoming emergencies.

• Predicting sports strategies

Sports coaches use the theory of probability to predict players’ actions in games such as chess.

• Aids in developing strategies for selling products

Product selling companies also use probability to sell their products. The application of these theories helps in getting premium prices for products.

There are also many other important uses of probability in daily life and mathematics.

Probability is a very important topic in both mathematics and real life. This chapter is mostly based on concepts and theories. Therefore, to understand the concept of probability and its application in terms of formulae and numbers, first, it is important to get a strong grip of the theories and NCERT Solutions for Class 9 Maths Chapter 15.

Through these NCERT Solutions for Class 9 Maths, we have covered the important aspects of probability such as its terms, types, and applications. The practice questions with solutions can help students in testing their newfound knowledge.