These Class 9 Maths Notes and Chapter 7 The Mathematics of Maybe Introduction to Probability Class 9 Ganita Manjari Notes are designed according to the latest CBSE syllabus.
Class 9 Maths Chapter 7 The Mathematics of Maybe Introduction to Probability Notes
Class 9 Maths Ganita Manjari Chapter 7 Notes
Class 9 The Mathematics of Maybe Introduction to Probability Notes
→ Introduction: In our day today lives, we come across many situations when we talk about the predictions made. We predict the results of matches, some environmental phenomenon which can occur in coming days, farmers predict their harvests, etc.
We talk about the chances more often in our day to day lives. Probably I will go there, he will come, he will play, etc are the phrases which we use almost on daily basis.
Probability is a type of measurement, similar to how we measure quantities like length, area, or volume. Unlike measuring physical quantities, probability is used’to measure the likelihood of events. Specifically, it helps us express how confident or certain we are that a particular event will occur. If we are certain about an event to occur, the answer is nearby 1, while if we are uncertain enough the answer is nearby 0.
→ Probability:
Probability is a measurement of the likelihood of an event. It is expressed on a scale from 0 (impossible) to 1 (sure), representing the likelihood of an event occurring. The probability of an event E is denoted as P(E), where 0 < P(E) < 1.
Probability is the area of mathematics that studies randomness and how likely a specific outcome is to happen in a random situation. For example, when we roll a die, probability of any number through 6 is \(\frac{1}{6}\), because each is equally likely.
→ Experimental Probability:
Experimental probability is determined by conduction an experiment and is calculated as follows
Experimental Probability = \(=\frac{\text { Number of times the event occured }}{\text { Total number of trials }} .\)
Relative frequency is the term which is also used for the experimental probability. It helps us understand probability based on actual data, rather than theoretical predictions. It is especially useful in statistics and data analysis when we are working with observed outcomes.
For example, suppose we roll a die 80 times, and it lands on a 6 exactly 16 times.
Experimental probability of rolling a 6 is \(\frac{16}{80}=\frac{1}{5}\) = 0.20 or 20%. So, the relative frequency of rolling a 6 is \(\frac{16}{80}\) = 0.20.
→ Theoretical Probability:
This approach assumes that all possible outcomes are equally likely.
Theoretical Probability = \(\frac{\text { Number of favorable outcomes }}{\text { Total number of possible outcomes }}\)
→ Random Events:
Events for which we know the possible outcomes but not the exact outcome for the ongoing event are called Random Events. There is an element of chance or randomness involved every time such an event takes place.
Randomness refers to a situation or action where you cannot exactly predict what will happen. Although you may know all the possible outcomes; you cannot say which one will definitely occur.
For example, In tossing a coin we know all the possible outcomes which can come which are Head or Tail, in rolling a die we know that one of the numbers 1, 2, 3, 4, 5 or, 6. These observations are called events.
→ Subjective Probability:
Response to the prediction is based on different kinds of evidence that have been gathered. The answer to the questions based on the prediction may be subjected to the evidences gathered over the time and sometimes based on the experience of life. This is the subjective probability.
→ Sample Space:
In an experiment, the event is a result of the experiment and is called an outcome. The set of all possible outcomes in an experiment is called the sample space. The sample space may be listed within brackets, separated by commas as shown below.
For example, In tossing a coin once possible outcomes are a head or a tail. Which may be listed as S = {H,T}. In rolling a die the possible outcomes are 1, 2, 3, 4, 5, 6. Which may be listed as
S = {1, 2, 3, 4, 5, 6}.
The sample space is denoted by S, is the list of all possible outcomes. Following points must be kept in mind while writing a sample space for an experiment:
- The sample space S must include every possible outcome.
- No outcome should be listed more than once.
- The number of elements in the sample space is called the sample size and is denoted by n(S).
→ Events:
An event is any single possible result or combination of results that might happen when we perform a random action. It is like choosing particular outcomes from all the things that could possibly occur.
For example, in the experiment of tossing a coin twice, S = {(HH), (HT), (TH), (TT)}, then E1: getting a head; E2: getting both tails; E3:getting atleast one head, etc, are the events associated with this experiment.
Also, E1 = {(HT),(TH)} has outcomes or sample points or elements (HT) and (TH),
E2 = {(TT)} has only one outcome or sample point or element (TT).
We can write sample size as n(S) = 4, we can also extend this notation to include the number of elements of various events. For example, we can write n(E1) = 2, n(E2) = 1, and so on.
→ A Tree Diagram:
A tree diagram is a visual representation used to list all possible outcomes of a multi-step experiment. A multi-step experiment involves a series of independent trials. For example, tossing a coin two times, or rolling a dice three times are examples of multi-step experiments. Each branch of the tree represents a possible outcome, and branches split to show different paths for subsequent events.
A tree diagram is useful for
- Visualising multi-step experiments, where each path from start to end represents one complete outcome.
- Listing all outcomes of a sample space.
- Calculate probabilities of events related to random experiments.
→ We can write expression for the probability of an event as:
P(E) = \(\frac{n(\mathrm{E})}{n(\mathrm{~S})}\)