Probability: Overview, Questions, Preparation

Probability can be defined as measuring the likelihood of an event or occurrence from a finite set of possible outcomes. It is according to NCERT. However, one cannot predict all events with complete certainty. We can only predict the chance of the occurrence of any particular event. 

Probability of an Event

Let us assume that an event ‘E’ can occur in ‘r’ different ways out of ‘n’ number of probable events. In this case, the probability of the event ‘E’ occurring is expressed as:

P(E) = r/n

Similarly, the probability that the event ‘E’ will not occur is expressed as:

P(E’) = (n-r)/n or 1-(r/n)

E’ denotes the expression of the non-occurrence of the event.

Thus, we can say that: P(E) + P(E’) = 1

This means that the sum of all probabilities in a random experiment is equal to 1.

The formula for probability is simple. It is the ratio of the number of favourable outcomes to an event to the total number of possible events.

Probability P(E) = No. of favourable outcomes/Total no. of possible outcomes

Importance of Probability

In class 10, you can expect 2-3 questions from probability ranging from questions related to dice, cards, coins, a bag of balls, etc. However, it is also crucial in class 12 with new theorems such as the Baye’s Theorem and more unique techniques using permutations and combinations. The weightage of class 12 differs from 2-3 questions as well. 

The two central principles in maths are probability and statistics. Probability is a question of chance. In contrast, statistics are all about how we treat different data using different methods. It helps to represent complex data in a very simple and clear way. 

Statistics is a study of data processing, analysis, perception, presentation, and organization. It is a method of capturing and summarizing data. This has many uses, from small to large scale. If it is a survey of the world’s population or its economy, the numbers are used for all such data analysis.

Terms used for probability and statistics 

Various expressions are found in odds and mathematical principles, such as:

• Random Experiments 
• Sample Space
• Variables of randomness 
• Value Expected 
• Independence 
• Variance 
• Mean's 

Let us discuss some of these terms. 

Random Experiments 

The experiment, the outcome of which cannot be expected before it is observed, is considered a random experiment. E.g., if we randomly throw a dice, the outcome is unknown to us. We can get any output between 1 and 6. This experiment is also spontaneous. 

Sample Space 

A sampling space is a collection of all possible outcomes or results of a random experiment. Suppose, if we throw a dice randomly, then the sample space for this experiment will be all the possible outcomes of throwing a dice, such as; 

Space Sample = { 1,2,3,4,5,6} 

Independent Occurrences 

If the likelihood of occurrence of one event does not affect the probability of another event, all occurrences are assumed to be independent of each other. E.g., if you flip a coin and, at the same time, throw a dice, the probability of having a 'head' is independent of the probability of getting a 6 in the dice.

Probability And Statistics in Class 11 

This concept is taught under the chapter Statistics And Probability. You will learn about the concepts of statistics and probability. The weightage of the chapter is 12 marks in the final exam.

Question 1: Find the probability of getting a six on rolling dice once?

Solution:

Sample Space = {1, 2, 3, 4, 5, 6}

No of favourable events = 1

Total no. of outcomes = 6

Thus, P = ⅙ (Answer)

Question 2: Draw a random card from a deck of cards. What is the probability that the card drawn is an ace?

Solution:

A standard deck of cards has 52 cards in total.

Therefore, the total no. of outcomes = 52

No. of favourable outcomes = 4 (one for each suit- 4 suits namely Spade, Club, Hearts Diamonds)

Probability = No. of Favourable Outcomes/Total No. of Outcomes = 4/52= 2/26= 1/13 (Answer)

Question 3: What is the probability of getting two consecutive heads if we flip a regular coin twice?

Solution:

Sample Space: (H, H), (H, T),(T, T)

No. of favourable events: 1

Total outcomes: 3

Required probability: ⅓ (Answer)

Question 4:Take the mean and mode of the following data: 2, 3, 5, 6, 10, 6, 12, 6, 3, 4. 

Solution: 

Complete number: 10 

Sum of all numbers: 2+3+5+6+10+6+6+12+3+7=60 

Mean = (sum of all numbers)/(Total number of items) 

Mean = 60/10 = 6 

Again, Number 6 appears 3 times, so Mode = 6

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