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Probability

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Class 12 Maths Chapter 13
Probability
Important Questions

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Probability is a branch of mathematics concerned with the likelihood of different outcomes in various scenarios. It quantifies the chance of events happening based on a scale from 0 (impossibility) to 1 (certainty). Explore critical questions related to Chapter 13 on probability for Class 12 Mathematics.. Crafted specifically to aid students in preparing for the CBSE Class 12 probability Examination 2024–25, these questions cover vital aspects of the topic. This mathematical concept is used in various fields such as statistics, finance, gambling, science, and engineering to make predictions and informed decisions based on the likelihood of certain outcomes.

Introduction

Probability is a fundamental concept in mathematics and statistics that deals with the likelihood or chance of different outcomes in an uncertain event or situation. It is a way to quantify the uncertainty associated with various phenomenon.probability is crucial for interpreting data and making decisions under uncertainty. It provides a mathematical framework for predicting the likelihood of future events based on past data or inherent randomness in natural phenomena.This introduction covers the basics of probability, setting the stage for more detailed study in areas like probability distributions, statistical inference, and stochastic processes.

What are Probability?

Probability is a branch of mathematics that deals with calculating the likelihood of a given event’s occurrence, which is expressed as a number between 0 and 1. It quantifies the chance that a specific event will occur in a set of conditions or a particular experiment. This measure, or probability, is expressed as a number between 0 and 1.
  • An event with a probability of 0 is an impossibility.
  • An event with a probability of 1 is a certainty.

Class 12 Probability Important Questions and Answers

Q1. A bag contains 8 black and 5 blue balls; three balls are drawn at random without replacements. What is the probability that all drawn balls are black colours?
Options
(a) 0.122
(b) 0.195
(c) 0.132
(d) 0.022

Ans. (b) 0.195
Explanation:
1^{st} black ball drawn at random the probability = \dfrac{8}{13} \\[4.5 bp] 2^{nd} black ball drawn at random the probability = \dfrac{7}{12} \\[4.5 bp] 3^{rd} black ball drawn at random the probability = \dfrac{6}{11} \\[4.5 bp] \text{P =} \left(\dfrac{8}{13}\right) × \left(\dfrac{7}{12}\right) × \left(\dfrac{6}{11}\right)\\[4.5 bp] \text{So the total probability = }\dfrac{28}{(13×11)}\\[4.5 bp] \text{= 0.195 }

Q2. Nine cards numbered from 1 to 9 are placed in a pack, mixed up thoroughly then one card is drawn randomly. If it is known that the number on each drawn card is more than 6. What is the probability that it is an even number?
Options
(a) \dfrac{1}{2} \\[4.5 bp](b) \dfrac{3}{4} \\[4.5 bp](c) 1
(d) \dfrac{1}{12}

Ans. (c)
Explanation:
A = {2, 4, 6, 8}
B = {8}
\\ \text{P(A) }= \dfrac{4}{9}, \text{ P(B) } = \dfrac{1}{9},\text{ A ∩ B = {8}, P(A ∩ B) }= \dfrac{1}{9}\\[4.5 bp] P\left(\dfrac{A}{B}\right) = P\dfrac{\text{(A ∩ B)}}{\text{P(B)}} = \dfrac{\left(\dfrac{1}{9} \right)}{\left(\dfrac{1}{9}\right)} = 1

Q3. Let A and B be two events such that
\text{P(A) } =\dfrac{3}{8} \text{and P(B) }= \dfrac{2}{5}. \\ Find P(A and B), if A and B are independent events.

Explanation:
Let A and B be two independent events. Then,
P(A ∩ B) = P(A) × P(B) = \left(\dfrac{3}{8}\right) × \left(\dfrac{2}{5}\right) = \left(\dfrac{6}{40}\right) = \dfrac{3}{20}.

Q4. If A and B are two events such that
\text{P(B) }= \dfrac{3}{5}, \text{P(A|B)} = \dfrac{1}{2} \\[4.5 bp] \text{and P(A ∪ B)} = \dfrac{4}{5}, \\ then find the value of P(A).

Explanation:
Here, P(B) = \dfrac{3}{5}, \text{P(A|B)} = \dfrac{1}{2}\text{ and P(A ∪ B) }= \dfrac{4}{5}\\[4.5 bp] \text{P(A|B) = } \dfrac{P(A ∪ B)}{P(B)}\\[4.5 bp] ⇒ \dfrac{1}{2} = \dfrac{P(A ∪ B)}{P(B)}\\[4.5 bp] ⇒ P(A ∩ B) = \left(\dfrac{3}{5}\right) x \left(\dfrac{1}{2}\right) = \dfrac{3}{10} \\[4.5 bp] and P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ \dfrac{4}{5} = \text{P(A) }+ \dfrac{3}{5} - \dfrac{3}{10} \\[4.5 bp] \text{P(A) =} \dfrac{4}{5} - \dfrac{3}{5} + \dfrac{3}{10}\\[4.5 bp] = \dfrac{(8 - 6 + 3)}{10}\\[4.5 bp]=\dfrac{1}{2} .

Q5. A problem in Mathematics is given to three students whose chances of solving it are
\dfrac{1}{2},\dfrac{1}{3}, \dfrac{1}{4} respectively. If the events of their solving the problem are independent then find the probability that the problem will be solved.

Explanation:
Let A, B, C be the respective events of solving the problem.

Then, \text{ P(A) }= \dfrac{1}{2} ,\text{ P(B) = } \dfrac{1}{3} \text{ and P(C) =}\dfrac{1}{4}.\\ Here, A, B, C are independent events.

Problem is solved if at least one of them solves the problem.

Required probability = P(A∪B∪C)
= 1 - P(A’) P(B’) P(C’)
= 1- \left(1 - \dfrac{1}{2}\right)\left(1 - \dfrac{1}{3}\right)\left(1 - \dfrac{1}{4}\right)\\[4.5 bp] = 1- \left(\dfrac{1}{2}×\dfrac{2}{3}×\dfrac{3}{4}\right)\\[4.5 bp] = 1- \dfrac{1}{4} = \dfrac{3}{4}.

CBSE Class 12 Maths Chapter wise Important Questions

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Conclusion

Probability is a vital and versatile field of mathematics that plays a crucial role in numerous aspects of daily life and various professional fields. Its principles are integral to various scientific, economic, and practical applications, underscoring its significance in both education and professional practice. Exploring CBSE class 12 maths probability important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics. oswal.io provide class 12 Probability important questions and answers.

Frequently Asked Questions

Ans: Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Ans: Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes in an experiment.
Ans: Independent events are those where the occurrence of one does not affect the probability of the other. Dependent events are when the occurrence of one event affects the probability of another.
Ans: Theoretical probability is based on the assumption that all outcomes are equally likely, whereas empirical probability is calculated based on historical data or experiments.
Ans: No, probability values always range from 0 to 1. A probability greater than 1 or less than 0 is not valid.
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