Probability Class 10th Notes - Free NCERT Class 10 Maths Chapter 15 Notes - Download PDF

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NCERT Class 10 Maths Chapter 14 Notes: Probability

Probability is defined as a measure of the possibility or chance that a particular event will occur. It assesses uncertainty and is used to determine the outcomes in various situations, from everyday life to scientific research.

Probability is expressed as a number between 0 and 1, which implies 0 means an event is impossible (e.g., rolling an 8 on a standard dice) and 1 means an event is certain (e.g., the sun rising tomorrow). In other terms, it is also represented as a percentage, where 100% means a specific event and 0% means an impossible event. A probability between 0 and 1 represents the degree of possibility of an event occurring. Probability of an Event E is represented by P(E).

Experimental Probability

Experimental Probability is founded on real-world observations and experimentation. It applies to any occurrence related to an experiment that is carried out numerous times.

The Experimental probability is calculated as follows:

P(E) = $ \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}}$

Theoretical Probability

Theoretical Probability is calculated by mathematical methods instead of experimental research or without conducting experiments.

For an event $E$, the theoretical probability is calculated as follows:

$P(E)$ = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Examples of Probability

1. Tossing of a Coin: When a coin is tossed, the probability of getting heads is:
   
   $P(E)$ = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
   
   Here, favourable outcomes are 1 (because in a coin, heads is one in count)
   
   And, the total number of possible outcomes is 2 (one is heads and the other is tails)
   
   So, P(Heads) = $\frac{1}{2}$ = $0.5$
2. Rolling a Dice: When a dice is rolled, the probability of getting 4 is:
   
   In this case, favourable outcomes are 1 (because on a dice, 4 comes only once)
   
   And, the total number of possible outcomes is 6 (there are 6 numbers on a dice)
   
   So, P(4) = $\frac{1}{6} = $0.1666$
3. Drawing a Black Card from a Deck: There are 52 cards in a deck, 26 of which are black. The probability of drawing a black card is:
   
   P(Black Card) = $\frac{26}{52} = \frac{1}{2} = 0.5 $

People use probability for multiple purposes, including statistics and gaming as well as finance and artificial intelligence, for deciding and predicting future risks.

Key Points

Some key points regarding the probability of Class 10 Maths Chapter 14 are:

• The expression of probability exists between 0 and 1. It means:
  0 ≤ P(E) ≤ 1 for any Event E
• Any event with a probability value of 0 is completely impossible.
• When an event carries a probability value of 1, it indicates a certain occurrence.
• When an event stands between 0 and 1, it demonstrates how probable the event is to happen.
• An event having only one outcome of the experiment is called an elementary event. Like, in the case of tossing a coin, there is only one outcome, either heads or tails.
• The sum of the probabilities of all the elementary events of an experiment is 1. Take an example of tossing a coin:
  P(heads) = 0.5
  P(tails) = 0.5
  They are both elementary events.
  So, P(heads) + P(tails) = 0.5 + 0.5 = 1

Key Terms in Probability

Given below are some important terms regarding the probability of Class 10 Maths Chapter 14:

Experiment:

A process generates various possible end results (e.g., rolling a die).

Sample Space (S):

The set of all possible outcomes of an experiment (e.g., {1, 2, 3, 4, 5, 6} for a die roll) is called as Sample Space.

For example,
A coin is tossed, and it is called an Event.
E1 = Having a head (H) on the upper face when tossed.
E2 = Having a tail (T) on the upper face when tossed.
So, in this case, the possible outcomes are Heads or Tails
Therefore, S = {H, T}

Event (E):

A subset of possible experimental outcomes which shows particular outcomes or collections of outcomes (e.g., rolling an even number).

Probability of an Event [P(E)]:

A measure of the likelihood of an event occurring, calculated as:

$ P(E)= \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Complementary Events:

The probability of an event not occurring is called the complementary event.

$ P(\overline{E}) = 1 – P(E) $

The event $\overline{E}$, representing ‘not E’, is called the complement of the event E. It means $\overline{E}$ and E are complementary events.

For example, Flipping of a Coin, where ‘getting a tail’ complemented the event of ‘getting a head’.

Elementary Events:

An event having only one outcome of the experiment is called an Elementary Event.

For example, when rolling a six-sided die, the sample space is:

S = {1, 2, 3, 4, 5, 6}

In this case, each outcome (1, 2, 3, 4, 5, or 6) is an elementary event because it consists of a single outcome.

Like,
Getting a 1 → E = {1}
Getting a 4 → E = {4}

Each of these events has a probability of: P(E) = $\frac{1}{6}$

Impossible Event:

An impossible event describes any outcome which will never occur within a particular experiment. It holds true that every impossible event will have a probability value of zero.

For example, the probability of getting a number 8 in a single throw of a die is 0. As 8 can never be the outcome.

Sure Event:

A sure event stands as an event which necessarily occurs throughout a particular experiment. A sure event (certain event) holds a fixed probability value of one.

For example, the probability of getting a number less than 7 in a single throw of a die is 1 because it is certain that when a die is thrown once, we will always receive a number less than seven. After all, each face of the die is marked with a number less than seven. Thus, there are six alternative outcomes, which is the same number of favourable options.

The comprehension of probability remains essential for satisfying decision-making processes when uncertainty exists in domains of finance, games, science and technology. Statistical inference, along with machine learning, stands upon probability as its basis, therefore making probability an essential mathematical concept linked to various domains.

Probability: Previous Year Question and Answer

Given below are some previous year question answers of various examinations from the NCERT class 10 chapter 14, Probability:

Question 1: Two dice are rolled together. The probability of getting an outcome (a, b) such that $b=2a$, is

Solution:
Total outcomes when two dice are rolled = $6 \times 6 = 36$.
Favourable outcomes $(a, b)$ where $b = 2a$:
If $a=1, b=2 \implies (1, 2)$
If $a=2, b=4 \implies (2, 4)$
If $a=3, b=6 \implies (3, 6)$
Favourable outcomes = 3
Probability = $\frac{\text{Favourable outcomes}}{\text{Total outcomes}} = \frac{3}{36} = \frac{1}{12}$
Hence, the correct answer is $\frac{1}{12}$.

Question 2: Three coins are tossed together. The probability that at least one head comes up is

Solution:
Total outcomes $=\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}$

Favourable outcomes $ =\{HHH, HHT, HTH, THH, HTT, THT, TTH\}$

Probability $=\frac{\text{Favourable Outcomes}}{\text{Total outcomes}}=\frac{7}{8}$

Hence, the correct answer is $\frac{7}{8}$.

Question 3: A box contains 120 discs, which are numbered from 1 to 120. If one disc is drawn at random from the box, find the probability that the number is a perfect square.

Solution:

Total discs = 120 (numbered 1 to 120).

So, the perfect squares are:

$1,4,9,16,25,36,49,64,81,100 $

Therefore, the total number of favourable outcomes = 10

Probability $= \frac{\text{No. of favourable outcomes}}{\text{Total number of outcomes}}$

Probability (perfect square) = $\frac{\text{Number of perfect squares}}{\text{Total number of discs}}$

$= \frac{10}{120}$

$= \frac{1}{12}$

Hence, the correct answer is $\frac{1}{12}$.

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