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Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If p is the probability that this perfect square is an odd number, then the value of 14p is _____
Correct answer is '8.00'. Can you explain this answer?
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Introduction:
We are given two fair dice, each with faces numbered 1, 2, 3, 4, 5, and 6. The dice are rolled together, and we observe the sum of the numbers on the faces. This process continues until the sum is either a prime number or a perfect square. We need to find the probability that the first perfect square we encounter is an odd number.
Solution:
Let's analyze the possible outcomes of rolling two dice together. The sum of the numbers on the faces can range from 2 (when both dice show 1) to 12 (when both dice show 6).
Possible Sums:
Here are the possible sums and their frequencies:
- Sum 2: 1 way (1, 1)
- Sum 3: 2 ways (1, 2) and (2, 1)
- Sum 4: 3 ways (1, 3), (2, 2), and (3, 1)
- Sum 5: 4 ways (1, 4), (2, 3), (3, 2), and (4, 1)
- Sum 6: 5 ways (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1)
- Sum 7: 6 ways (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)
- Sum 8: 5 ways (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2)
- Sum 9: 4 ways (3, 6), (4, 5), (5, 4), and (6, 3)
- Sum 10: 3 ways (4, 6), (5, 5), and (6, 4)
- Sum 11: 2 ways (5, 6) and (6, 5)
- Sum 12: 1 way (6, 6)
Prime Numbers:
Now let's identify the prime numbers among these sums: 2, 3, 5, 7, 11.
Perfect Squares:
The perfect squares among these sums are: 4, 9.
Calculating the Probability:
Since the process continues until we encounter a perfect square or a prime number, we need to calculate the probability of encountering a perfect square before a prime number.
The probability of encountering a perfect square is 2/12 = 1/6, as there are 2 perfect squares out of 12 possible sums.
The probability of encountering a prime number is 5/12, as there are 5 prime numbers out of 12 possible sums.
Therefore, the probability of encountering a perfect square before a prime number is (1/6) / (1/6 + 5/12) = 2/7.
The probability that the first perfect square encountered is an odd number is 1/2 since there is only one odd perfect square (9) out of the two possible perfect squares (4 and 9).
H
We are given two fair dice, each with faces numbered 1, 2, 3, 4, 5, and 6. The dice are rolled together, and we observe the sum of the numbers on the faces. This process continues until the sum is either a prime number or a perfect square. We need to find the probability that the first perfect square we encounter is an odd number.
Solution:
Let's analyze the possible outcomes of rolling two dice together. The sum of the numbers on the faces can range from 2 (when both dice show 1) to 12 (when both dice show 6).
Possible Sums:
Here are the possible sums and their frequencies:
- Sum 2: 1 way (1, 1)
- Sum 3: 2 ways (1, 2) and (2, 1)
- Sum 4: 3 ways (1, 3), (2, 2), and (3, 1)
- Sum 5: 4 ways (1, 4), (2, 3), (3, 2), and (4, 1)
- Sum 6: 5 ways (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1)
- Sum 7: 6 ways (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)
- Sum 8: 5 ways (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2)
- Sum 9: 4 ways (3, 6), (4, 5), (5, 4), and (6, 3)
- Sum 10: 3 ways (4, 6), (5, 5), and (6, 4)
- Sum 11: 2 ways (5, 6) and (6, 5)
- Sum 12: 1 way (6, 6)
Prime Numbers:
Now let's identify the prime numbers among these sums: 2, 3, 5, 7, 11.
Perfect Squares:
The perfect squares among these sums are: 4, 9.
Calculating the Probability:
Since the process continues until we encounter a perfect square or a prime number, we need to calculate the probability of encountering a perfect square before a prime number.
The probability of encountering a perfect square is 2/12 = 1/6, as there are 2 perfect squares out of 12 possible sums.
The probability of encountering a prime number is 5/12, as there are 5 prime numbers out of 12 possible sums.
Therefore, the probability of encountering a perfect square before a prime number is (1/6) / (1/6 + 5/12) = 2/7.
The probability that the first perfect square encountered is an odd number is 1/2 since there is only one odd perfect square (9) out of the two possible perfect squares (4 and 9).
H
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Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer?
Question Description
Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer?.
Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pis the probability that this perfect square is an odd number, then the value of 14pis _____Correct answer is '8.00'. Can you explain this answer?.
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