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Three numbers, ,a b c are chosen from (0, 4) . The probability that the sum of these three numbers is greater than 2 is? Use the concept of geometric probability. The answer is 1/24
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Three numbers, ,a b c are chosen from (0, 4) . The probability that th...
Understanding the Problem
To find the probability that the sum of three numbers a, b, and c, chosen from the interval (0, 4), exceeds 2, we will use geometric probability.
Defining the Region of Interest
1. Domain of Selection: Each number a, b, and c can take values in the interval (0, 4). Thus, we are considering the cube defined by 0 < a,="" b,="" c="" />< />
2. Total Volume Calculation: The total volume of this cube is:
- Volume = 4 * 4 * 4 = 64.
Identifying the Favorable Region
1. Inequality Condition: We need to find the volume of the region where a + b + c > 2.
2. Complementary Region: It's easier to find the complementary region where a + b + c ≤ 2.
Finding the Complement Region Volume
1. Setting Up: The inequality a + b + c ≤ 2 defines a tetrahedron in the first octant of the coordinate system.
2. Volume of the Tetrahedron: The vertices of this tetrahedron are (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2). The volume is calculated using the formula for the volume of a tetrahedron:
- Volume = (1/6) * base area * height.
- Here, base area = 1/2 * 2 * 2 = 2, and height = 2. Thus, total volume = (1/6) * 2 * 2 = 2/3.
Calculating the Desired Probability
1. Volume of Favorable Region: The volume where a + b + c > 2 is:
- Favorable Volume = Total Volume - Complement Volume = 64 - (2/3) = 64 * (3/3) - (2/3) = 192/3 - 2/3 = 190/3.
2. Probability Calculation: The probability that the sum is greater than 2 is:
- Probability = Favorable Volume / Total Volume = (190/3) / 64 = 190 / 192 = 95 / 96.
However, in the context of the question, an error occurred in the stated answer of 1/24. The correct analysis shows that the probability is actually 95/96.
To find the probability that the sum of three numbers a, b, and c, chosen from the interval (0, 4), exceeds 2, we will use geometric probability.
Defining the Region of Interest
1. Domain of Selection: Each number a, b, and c can take values in the interval (0, 4). Thus, we are considering the cube defined by 0 < a,="" b,="" c="" />< />
2. Total Volume Calculation: The total volume of this cube is:
- Volume = 4 * 4 * 4 = 64.
Identifying the Favorable Region
1. Inequality Condition: We need to find the volume of the region where a + b + c > 2.
2. Complementary Region: It's easier to find the complementary region where a + b + c ≤ 2.
Finding the Complement Region Volume
1. Setting Up: The inequality a + b + c ≤ 2 defines a tetrahedron in the first octant of the coordinate system.
2. Volume of the Tetrahedron: The vertices of this tetrahedron are (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2). The volume is calculated using the formula for the volume of a tetrahedron:
- Volume = (1/6) * base area * height.
- Here, base area = 1/2 * 2 * 2 = 2, and height = 2. Thus, total volume = (1/6) * 2 * 2 = 2/3.
Calculating the Desired Probability
1. Volume of Favorable Region: The volume where a + b + c > 2 is:
- Favorable Volume = Total Volume - Complement Volume = 64 - (2/3) = 64 * (3/3) - (2/3) = 192/3 - 2/3 = 190/3.
2. Probability Calculation: The probability that the sum is greater than 2 is:
- Probability = Favorable Volume / Total Volume = (190/3) / 64 = 190 / 192 = 95 / 96.
However, in the context of the question, an error occurred in the stated answer of 1/24. The correct analysis shows that the probability is actually 95/96.
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Three numbers, ,a b c are chosen from (0, 4) . The probability that the sum of these three numbers is greater than 2 is? Use the concept of geometric probability. The answer is 1/24?
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Three numbers, ,a b c are chosen from (0, 4) . The probability that the sum of these three numbers is greater than 2 is? Use the concept of geometric probability. The answer is 1/24? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Three numbers, ,a b c are chosen from (0, 4) . The probability that the sum of these three numbers is greater than 2 is? Use the concept of geometric probability. The answer is 1/24? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Three numbers, ,a b c are chosen from (0, 4) . The probability that the sum of these three numbers is greater than 2 is? Use the concept of geometric probability. The answer is 1/24?.
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