CBSE Class 10 Maths Notes Chapter 15 Probability is an excellent resource, for knowing all the concepts of a particular chapter in a crisp, and friendly manner. Our articles, help students learn in their language, with proper images, and solved examples for a better understanding of the concepts.
Chapter 15 of the NCERT Class 10 Maths is probability and covers various topics such as understanding experimental and theoretical probability, types of events, and elementary and non-elementary probability. These notes are designed to provide students with a comprehensive summary of the entire chapter and include all the essential topics, formulae, and concepts needed to succeed in their exams. Additionally, these notes are suitable for Probability Class 10 students.
Need of Studying Probability
We are using probability in day-to-day life, but we have never realized it. For example, throwing a die, you know that only one of the six numbers can always be shown by a dice, hence, the probability of getting a particular number on the dice is 1/6. Probability, has vast applications in the field of science, for example, scientists have never been able to find the position of an electron, because it is insanely small, so, we always define the position of an electron in terms of probability, i.e. there are 80% chances that an electron will be at that position, in an atom, this also forms the basis of quantum mechanics. Hence, the study of probability is important and it is taught in class 10.
What is Probability?
Probability tells the possibility of occurring an event. It can also be stated as the ratio of favorable outcomes and the total number of outcomes. Probability always lies in the range of 0 and 1, and can also be expressed in terms of the percentage. There are 2 different types of probabilities i.e. experimental probability and theoretical probability. In the 9th class, we studied experimental probability in detail, and in the 10th we will be studying theoretical probability.
Types of Probability
In Probability theory, it can broadly be classified into 2 types as follows:
- Experimental Probability
- Theoretical Probability
Let’s understand these types in detail.
Experimental Probability
In the 9th class, we studied experimental probability. Experimental probability is the probability, which tells the possible outcomes, after performing an experiment. For example, if you toss a coin 50 times, then we could have different outcomes, in repeating experiments i.e. for experiment 1, the number of heads = 23, and the number of tails = 27, for experiment 2, the number of heads = 26, and a number of tails = 24, and so on… Here, if we find the probability of getting a head, it will be nearer to 0.5, but not exactly will be equal to 0.5.
Below is the formula for calculating experimental probability:
Probabitlity = number of required outcomes after performing an experiment / total number of times experiment performed
Theoretical Probability
In this chapter, we will be studying theoretical probability in detail. Theoretical probability, tells how likely an event is going to occur, and in these events, we will only be assuming what can be the possible outcomes, but will not be performing experiments. For example, as we have seen above in the coin example, the experimental probability of getting a head/tail is near 0.5, but not exactly equal to 0.5, but theoretical probability, says that the probability of getting a head/tail is equal to 0.5, as head and tail are equally likely outcomes. Theoretical probability is also called classical probability.
Below is the formula for calculating theoretical probability:
P(E) = Number of trials in which an event occured/ total number of trials
Events and Outcomes
Before moving on to probability, we need to understand what are events and outcomes. An outcome is the result of an experiment, and this experiment is called an event that has possible outcomes. For example, choosing an Ace from a deck of 52 cards, is an event, with outcomes, diamond Ace, spade Ace, club Ace, and heart Ace. Now, we have different types of events and outcomes.
Types of Events
There are 5 different types of events i.e. sure events, impossible events, normal events, complementary events, and elementary events.
Sure Events
A sure event is an event that will always occur, irrespective of the other factors. For example, the probability of getting a number between 1 and 6, is 1. Hence, the probability of a sure event is always equal to 1.
P(Sure Event) = 1
Impossible Events
An impossible event is an event that can never happen. For example, the probability of getting a number greater than 6, is 0. Hence, the probability of an impossible event is always equal to 0.
P(Impossible Event) = 0
Normal Events
A normal event is an event, that can have any probability, i.e. between 0 and 1 inclusive. A sure event and an impossible event, are also normal events. For example, the probability of getting 1 on a die is 1/6.
0 <= P(E) <= 1
Complementary Events
A complementary event is an event, such that the sum of the probability of occurring an event and the probability of not occurring an event is equal to 1. For example, the probability of getting a number less than 3 on a die is, 2/6, and then the probability of not getting a number less than 3 is 4/6. Hence, the sum of their probabilities is equal to 1.
Elementary Events
An elementary event is an event, which has only one outcome. For example, the probability of getting the number 2 on a die is, 1/6, similarly, the probability of getting the number 1 on a die is 1/6, and similar for all the rest numbers….. The sum of probabilities of all elementary events is equal to 1.
P(E1) + P(E2) + P(E3) + … = 1
Types of Outcomes
There are two different types of outcomes i.e. equal likely outcome, and non-equally likely outcome.
Equally likely outcome
An equally likely outcome is the outcome that has, the same output and probability. For example, tossing a coin will give an equally likely outcome, i.e. head or tail, hence the probability of each of them is 1/2, rolling a die, will give an equally likely outcome i.e. 1, 2, 3, 4, 5, 6, hence the probability of each of them is 1/6.
Non-Equally likely outcome
A non-equally likely outcome is an outcome, that has uncertainty in its outcomes and also depends on the various other factors. For example, starting a car, and the car starts or does not start, this gives a non-equally likely outcome, because the car will always start unless it needs maintenance/service.
Cards Probability
The 52 playing cards in a deck are separated into four suits of hearts, diamonds, clubs, and spades.
Specific symbols are used to represent the four suits:
- Hearts: A red heart symbol (♥) represents the heart suit.
- Diamonds: A red diamond symbol (♦) represents the diamond suit.
- Clubs: A black club symbol (♣) represents the club suit.
- Spades: A black spade symbol (â™ ) represents the spade suit.
Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King are the 13 cards that make up each suit.
If we draw a card from a properly shuffled deck, it may happen in one of several ways:
- Drawing a specific card: In this case, a specific card, such as the Ace of Spades or the Queen of Hearts, is drawn from the deck. (P = 1/52)
- Drawing a card from a certain suit: In this scenario, a card from a particular suit, such as a heart or a diamond, is drawn. (P = 1/13)
- Drawing a card of a particular rank: The emphasis of this event is on drawing a card of a particular rank, such as a King or a 7. (P = 1/13)
- Drawing a face card: In this case, the King, Queen, and Jack of each suit are included in the face card. (P = 3/13)
- Drawing a red or black card: This event refers to drawing a card that is either red (hearts or diamonds) or black (clubs or spades). (P = 1/2)
Other than these events, there can be various events such as drawing a pair of cards, drawing a card of value greater than a specific number, etc.
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Practice Problems
Understanding only the theory part of this chapter is not enough. This chapter requires an understanding of different problem statements, and how to solve them. Most of the questions, in probability, are from coin selection, throwing a die, picking an item from a bag, and selecting cards. We will be solving questions on each topic for a better understanding of the chapter.
Problem 1: What is the probability of getting head and tail, when a coin is tossed twice?
Solution:
If a coin is tossed two times, then the possible number of outcomes are: HH, HT, TH, and TT.
So, there are 4 total possible outcomes,
As, we need to find the probability of getting head and tail, so the favourable outcomes are HT, and TH,
So, there are 2 favourable outcomes,
We know jthat,
P(E) = Number of favourable outcomes / Total number of outcomes,
⇒ P(E) = 2/4
⇒ P(E) = 0.5
Problem 2: If two die are rolled, at the same time. What is the probability of getting the same number on both dies?
Solution:
As, we are rolling two dies, so the possible number of outcomes are 6 x 6 = 36,
The above table shows the all possible outcomes,
Now, we need to find the pairs which has the same numbers i.e. (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6),
So, the favourable outcomes are 6,
We know that,
P(E) = Number of favourable outcomes/ Total number of outcomes,
⇒ P(E) = 6/36,
⇒ P(E) = 1/6
Problem 3: A bag contains 1 red ball, 2 green balls, and 2 white balls. What is the probability of not getting a white ball?
Solution:
Total number of balls = 5,
Number of red balls = 1, Number of green balls = 2, and Number of white balls = 2,
We know that,
,
P(Getting a white ball) + P(Not getting a white ball) = 1,
Thus, P(Not getting a white wball) = 1 – P(Getting a white ball), . . . (1)
P(Getting a whtie ball) = number of white balls / total number of balls,
Thus, P(Getting a white ball) = 2/5, . . . (2)
Put value of equation (2) in (1),
P(Not getting a white ball) = 1 – 2/5,
⇒ P(Not getting a white ball) = 3/5
Problem 4: Given a deck of 52 cards. Find the probability of getting a red Ace.
Solution:
Total number of cards = 52,
We have 26 red cards, and 26 white cards,
In 26 red cards, there are 13 diamond cards, and 13 heart cards, and each of them has 1 Ace.
Hence, total number of red ace = 2 (one diamond ace, and one heart ace),
We know that,
P(Event) = Favourable Outcomes / Total Outcomes,
⇒ P(Getting a red Ace) = number of red ace / total cards,
⇒ P(Getting a red Ace) = 2/52
⇒ P(Getting a red Ace) = 1/26
Problem 5: A die is rolled. What is the probability of getting a number less than 7? Also, find the complimentary probability of it.
Answer:
If a die is rolled we can get either of the 6 possibilities = {1, 2, 3, 4, 5, 6},
As, all numbers are less than 7,
Hence, our sample space = 6,
P(Getting a number less than 7) = Sample space / total number of possibilities,
⇒ P(Gettin a number less than 7) = 6/ 6 = 1,
From Complimentary events, we know that,
P(E) + P(Not E) = 1,
⇒ P(Not getting a number less than 7) = 1 – P(Getting a number less than 7),
⇒ P(Not getting a number less than 7) = 0
Thus, probability of getting a number less than 7 while rolling a die is 0 i.e., impossible event.
FAQs on NCERT Notes for Class 10 Maths Chapter 15 Probability
1. What is Probability in Mathematics?
Probability in mathematics is defined as the ratio of the favourable outcomes to the total number of outcomes.
2. How do you calculate the probability of an event?
We can calculate the probability using the mathematical formula:
P(E) = Favorable Outcomes / Total Outcomes
3. What Topics are Covered in Probability Class 10?
Some topics covered in the class 10 chapter probability are:
- Introduction to Probability
- Random Experiments
- Sample Space and Events
- Theoretical Probability
- Empirical Probability
4. What is Theoretical Probability?
Theoretical probability is the probability which tells how likely an event can occur, without performing the actual experiments.
5. What is a Sure Event?
A sure event is an event which has the P(E) = 1. For example, getting head or tail on tossing a coin is a sure event.
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