14. Probability
рдЗрд╕ рдкреЗрдЬ рдкрд░ рдЖрдкрдХреЛ 14. Probability рдХреЗ рд╕рднреА рдорд╣рддреНрд╡рдкреВрд░реНрдг рдмрд╣реБрд╡рд┐рдХрд▓реНрдкреАрдп рдкреНрд░рд╢реНрди (MCQ), рдкрд░реАрдХреНрд╖рд╛ рдЙрдкрдпреЛрдЧреА рдиреЛрдЯреНрд╕ рдФрд░ рдСрдирд▓рд╛рдЗрди рдХреНрд╡рд┐рдЬрд╝ рдорд┐рд▓реЗрдВрдЧреЗред рдЕрдкрдиреЗ рдЬреНрдЮрд╛рди рдХрд╛ рдкрд░реАрдХреНрд╖рдг рдХрд░рдиреЗ рдФрд░ рдмреЛрд░реНрдб/рдкреНрд░рддрд┐рдпреЛрдЧреА рдкрд░реАрдХреНрд╖рд╛ рдореЗрдВ рд╢рдд-рдкреНрд░рддрд┐рд╢рдд рдЕрдВрдХ рдкреНрд░рд╛рдкреНрдд рдХрд░рдиреЗ рдХреЗ рд▓рд┐рдП рдиреАрдЪреЗ рджрд┐рдП рдЧрдП рдореЙрдХ рдЯреЗрд╕реНрдЯ рдореЗрдВ рдЕрд╡рд╢реНрдп рднрд╛рдЧ рд▓реЗрдВред
ЁЯСЙ рдЕрдзреНрдпрд╛рдп рдХрд╛ рд╕рд╛рд░рд╛рдВрд╢ (Summary)
Dear students, in this final chapter of your mathematics curriculum, we will explore the intriguing world of Probability. Probability is the mathematical study of uncertainty. While we often think of the future as unpredictable, probability allows us to quantify the likelihood of various events occurring. From the simple flip of a coin to complex decision-making in insurance, weather forecasting, and stock market analysis, the principles of probability are used everywhere. This chapter is essential for your NCERT and board exam preparation as it helps you build a logical approach to reasoning and data analysis.
At its core, the probability of an event is defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes. For any event E, the probability P(E) always lies between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 indicates a certain event. You will learn to calculate probabilities for classic experiments such as tossing fair coins, rolling standard dice, and drawing cards from a well-shuffled deck. These foundational experiments teach you to systematically list all possible outcomes, which is the most critical step in any probability problem.
We will also look at the concept of complementary events. The sum of the probability of an event happening and the probability of it not happening is always exactly 1. In other words, P(E) + P(not E) = 1. This simple yet powerful rule allows us to calculate difficult probabilities by finding the probability of the opposite event and subtracting it from 1. This approach often simplifies calculations significantly, especially when you are asked to find the probability that something 'does not' happen.
Throughout this chapter, remember that probability is not just about memorizing formulas; it is about clearly defining what constitutes a 'favorable outcome' and 'total outcomes' in a given scenario. For example, when dealing with a deck of 52 playing cards, you must be familiar with the composition of the deck, including the suits, face cards, and numerical cards, to correctly identify the number of favorable outcomes. Real-world applications are vast; for instance, quality control engineers in manufacturing plants use probability to estimate the likelihood of a product being defective, helping companies maintain high standards.
As you practice these problems, always ensure that your outcomes are 'equally likely' before applying the basic probability formula. If the outcomes are biased or not equally likely, the simple ratio formula will not hold. With consistent practice and careful observation, you will master the art of predicting the likelihood of events with precision and confidence.
рд╡рд┐рд╢реЗрд╖рддрд╛: рдпрд╣ рдПрдХ 3-Level рдЖрдзрд╛рд░рд┐рдд рдпреВрдиреАрдХ рдХреНрд╡рд┐рдЬрд╝ рд╣реИред рдЕрдЧрд▓реЗ рд╕реНрддрд░ (Level 2 рдФрд░ 3) рдХреЛ рдЕрдирд▓реЙрдХ рдХрд░рдиреЗ рдХреЗ рд▓рд┐рдП рдЫрд╛рддреНрд░реЛрдВ рдХреЛ рд╡рд░реНрддрдорд╛рди рд╕реНрддрд░ рдореЗрдВ рдХрдо рд╕реЗ рдХрдо 70% рдЕрдВрдХ рд╕реНрдХреЛрд░ рдХрд░рдиреЗ рд╣реЛрдВрдЧреЗ, рдЬреЛ рдЙрдирдХреА рдмреЛрд░реНрдб рдкрд░реАрдХреНрд╖рд╛ рдХреА рддреИрдпрд╛рд░реА рдХреЛ 100% рдордЬрдмреВрдд рдмрдирд╛рддрд╛ рд╣реИред
рдорд╣рддреНрд╡рдкреВрд░реНрдг рдкреНрд░рд╢реНрди (Important Questions)
PDF рдбрд╛рдЙрдирд▓реЛрдб рдХрд░реЗрдВрдЕрдзреНрдпрд╛рдп рдХреЗ рд╕рднреА рдкреНрд░рд╢реНрди (Revision Notes)
*рдиреЛрдЯ: рдпрд╣ рд╕реЗрдХреНрд╢рди рд░рд┐рд╡реАрдЬрди рдХреЗ рд▓рд┐рдП рд╣реИред рдЕрдкрдирд╛ рдЬреНрдЮрд╛рди рдкрд░рдЦрдиреЗ рдХреЗ рд▓рд┐рдП рдКрдкрд░ рджрд┐рдП рдЧрдП рдХреНрд╡рд┐рдЬрд╝ рдореЗрдВ рднрд╛рдЧ рд▓реЗрдВред
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The probability of an impossible event is?
A) 1 | B) 0 | C) 0.5 | D) 0.25 -
The probability of a sure event is?
A) 0 | B) 0.5 | C) 1 | D) 10 -
If P(E) is the probability of an event E, then the range of P(E) is?
A) P(E) < 0 | B) P(E) > 1 | C) 0 <= P(E) <= 1 | D) P(E) = 1 -
When a fair coin is tossed, the probability of getting a head is?
A) 1 | B) 0 | C) 0.5 | D) 0.25 -
A die is thrown once. What is the probability of getting a number less than 7?
A) 0 | B) 1 | C) 0.5 | D) 1/6 -
In a deck of 52 playing cards, how many hearts are there?
A) 4 | B) 13 | C) 26 | D) 12 -
What is the probability of getting an even number in a single throw of a die?
A) 1/2 | B) 1/3 | C) 1/4 | D) 1 -
If P(E) = 0.05, then P(not E) is?
A) 0.5 | B) 0.95 | C) 0.9 | D) 0.05 -
Which of the following can never be the probability of an event?
A) 2/3 | B) 0.7 | C) 1.5 | D) 15% -
The sum of probabilities of all elementary events of an experiment is?
A) 0 | B) 0.5 | C) 1 | D) 2 -
A card is drawn from a well-shuffled deck of 52 cards. The probability of getting a red king is?
A) 1/52 | B) 1/26 | C) 1/13 | D) 2/13 -
A die is thrown twice. What is the probability that the sum of the numbers appearing on them is 10?
A) 1/12 | B) 1/9 | C) 1/6 | D) 1/36 -
One card is drawn from a pack of 52 cards. The probability of getting a face card is?
A) 3/13 | B) 4/13 | C) 1/13 | D) 2/13 -
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, then the number of blue balls is?
A) 5 | B) 10 | C) 15 | D) 20 -
Two dice are thrown simultaneously. What is the probability of getting a doublet?
A) 1/6 | B) 1/3 | C) 1/2 | D) 5/36 -
A letter is chosen at random from the letters of the word PROBABILITY. The probability that it is a vowel is?
A) 2/11 | B) 3/11 | C) 4/11 | D) 5/11 -
A bag contains cards numbered 1 to 25. A card is drawn at random. The probability that the number is divisible by 5 is?
A) 1/5 | B) 2/25 | C) 3/25 | D) 4/25 -
If a card is drawn from a pack of 52 cards, the probability of getting a black queen is?
A) 1/52 | B) 1/26 | C) 1/13 | D) 1/4 -
Two coins are tossed together. The probability of getting at most one head is?
A) 1/4 | B) 3/4 | C) 1/2 | D) 1 -
A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random, the probability that it bears a two-digit number is?
A) 81/90 | B) 9/10 | C) 1/10 | D) 9/90 -
A die is thrown twice. The probability of getting a sum less than 5 is?
A) 1/6 | B) 1/9 | C) 1/12 | D) 1/18 -
If three coins are tossed simultaneously, the probability of getting exactly two heads is?
A) 1/8 | B) 3/8 | C) 1/2 | D) 3/4 -
A game consists of tossing a one rupee coin 3 times and noting its outcome each time. If all tosses give the same result, Hanif wins, otherwise he loses. The probability that Hanif loses is?
A) 3/4 | B) 1/4 | C) 1/2 | D) 7/8 -
A box contains 5 red, 4 green and 7 white marbles. One marble is drawn at random. The probability that the marble is not red is?
A) 5/16 | B) 11/16 | C) 7/16 | D) 4/16 -
Two dice are thrown simultaneously. The probability of getting a sum greater than 9 is?
A) 1/6 | B) 1/4 | C) 1/9 | D) 5/36 -
A card is drawn from a pack of 52 cards. The probability that it is a spade or an ace is?
A) 4/13 | B) 16/52 | C) 17/52 | D) 13/52 -
A child has a die whose 6 faces show the letters as follows: A, B, C, A, B, A. If the die is thrown once, the probability of getting A is?
A) 1/2 | B) 1/3 | C) 2/3 | D) 5/6 -
A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The probability that she will not buy is?
A) 5/36 | B) 31/36 | C) 20/144 | D) 124/144 -
Two dice are thrown simultaneously. The probability that the product of the numbers is prime is?
A) 1/6 | B) 1/3 | C) 1/9 | D) 5/36 -
A bag contains 5 white, 7 red, 4 black and 2 blue balls. A ball is drawn at random. The probability that the drawn ball is white or blue is?
A) 1/2 | B) 7/18 | C) 1/9 | D) 7/9
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