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Chapter Analysis
Intermediate24 pages • EnglishQuick Summary
This chapter on probability covers the fundamental concepts and axioms of probability, including sample spaces, events, mutually exclusive and exhaustive events, and the computation of probabilities for different outcomes. It uses examples such as drawing cards, tossing coins, and examining defective items to introduce and explain the axiomatic approach to probability. The chapter illustrates the calculation of probabilities in various scenarios and concludes with exercises to practice the concepts learned.
Key Topics
- •Sample Space
- •Events and Outcomes
- •Mutually Exclusive Events
- •Exhaustive Events
- •Axiomatic Approach of Probability
- •Probabilities of Events
- •Venn Diagram Representation
- •Calculation of Combined Probabilities
Learning Objectives
- ✓Understand the concept of a sample space and events.
- ✓Differentiate between mutually exclusive and exhaustive events.
- ✓Apply the axiomatic approach to solve probability problems.
- ✓Learn to calculate probabilities using various formulas.
- ✓Construct and analyze probability scenarios using Venn diagrams.
- ✓Develop problem-solving skills in probability.
Questions in Chapter
A die is rolled. Let E be the event 'die shows 4' and F be the event 'die shows even number'. Are E and F mutually exclusive?
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2. A die is thrown. Describe the following events: (i) A: a number less than 7 (ii) B: a number greater than 7 (iii) C: a multiple of 3 (iv) D: a number less than 4 (v) E: an even number greater than 4 (vi) F: a number not less than 3
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3. An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: A: the sum is greater than 8, B: 2 occurs on either die, C: the sum is at least 7 and a multiple of 3.
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4. Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show, and D denote the event ‘a head shows on the first coin”.
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5. Three coins are tossed. Describe (i) Two events which are mutually exclusive. (ii) Three events which are mutually exclusive and exhaustive. (iii) Two events, which are not mutually exclusive. (iv) Two events which are mutually exclusive but not exhaustive. (v) Three events which are mutually exclusive but not exhaustive.
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Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.
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6. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35. Find (i) P(A ∪ B) (ii) P(A´ ∩ B´) (iii) P(A ∩ B´) (iv) P(B ∩ A´)
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A committee of two persons is selected from two men and two women. What is the probability that the committee will have (a) no man? (b) one man? (c) two men?
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Additional Practice Questions
If a bag contains 5 red balls, 3 green balls, and 2 blue balls, what is the probability of drawing a blue ball?
easyAnswer: Total number of balls = 5 + 3 + 2 = 10. The number of blue balls = 2. Therefore, the probability of drawing a blue ball = 2/10 = 1/5.
A fair six-sided die is rolled twice. What is the probability that the sum of the numbers is 9?
mediumAnswer: The possible outcomes pairs with a sum of 9 are (3,6), (4,5), (5,4), (6,3). Each pair has a probability of (1/6)*(1/6) = 1/36. Thus, total probability = 4*(1/36) = 1/9.
In a quiz, a contestant is awarded 10 points for each correct answer but loses 5 points for each incorrect answer. If a contestant answers 12 questions with 8 correct and 4 incorrect, what is the total score?
easyAnswer: Total score = (8 correct * 10) - (4 incorrect * 5) = 80 - 20 = 60 points.
What is the probability of getting at least one 6 in two roll of a fair six-sided die?
mediumAnswer: The probability of not getting a 6 in a single roll is 5/6. Thus, the probability of not getting a 6 in two rolls is (5/6)*(5/6) = 25/36. Therefore, the probability of getting at least one 6 = 1 - 25/36 = 11/36.
In a class of 30 students, 18 like mathematics, and 15 like science. If 10 students like both subjects, find the probability that a student selected randomly likes at least one of these subjects.
mediumAnswer: The probability of liking at least one subject = P(Mathematics) + P(Science) - P(Both). So, the number of students liking at least one subject is 18 + 15 - 10 = 23. Probability = 23/30.