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Two integers are selected at random from the set {1, 2,...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :
- a)2/5
- b)1/2
- c)3/5
- d)7/10
Correct answer is option 'A'. Can you explain this answer?
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Two integers are selected at random from the set {1, 2,...., 11}. Give...
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Two integers are selected at random from the set {1, 2,...., 11}. Give...
Given:
Two integers are selected at random from the set {1, 2,...., 11}.
To find:
The conditional probability that both the numbers are even, given that the sum of selected numbers is even.
Solution:
Step 1: Total number of ways to select two numbers:
The total number of ways to select two numbers from the set {1, 2,...., 11} is given by the combination formula, which is C(11, 2) = 55.
Step 2: Total number of ways to select two even numbers:
Out of the numbers from 1 to 11, there are 6 even numbers (2, 4, 6, 8, 10, 12).
The total number of ways to select two even numbers from these 6 even numbers is given by the combination formula, which is C(6, 2) = 15.
Step 3: Total number of ways to select two odd numbers:
Out of the numbers from 1 to 11, there are 5 odd numbers (1, 3, 5, 7, 9).
The total number of ways to select two odd numbers from these 5 odd numbers is given by the combination formula, which is C(5, 2) = 10.
Step 4: Total number of ways to select one even and one odd number:
Out of the numbers from 1 to 11, there are 6 even numbers (2, 4, 6, 8, 10, 12) and 5 odd numbers (1, 3, 5, 7, 9).
The total number of ways to select one even and one odd number is given by the product of the number of ways to select one even number and the number of ways to select one odd number, which is 6 * 5 = 30.
Step 5: Probability that both numbers are even, given that the sum is even:
The sum of two numbers will be even if both numbers are either even or odd. So, the favorable outcomes for both numbers being even is selecting two even numbers.
The probability that both numbers are even, given that the sum is even, is given by the ratio of the number of ways to select two even numbers to the total number of ways to select two numbers.
Therefore, the required probability is 15/30 = 1/2.
So, the correct answer is option 'B' - 1/2.
Two integers are selected at random from the set {1, 2,...., 11}.
To find:
The conditional probability that both the numbers are even, given that the sum of selected numbers is even.
Solution:
Step 1: Total number of ways to select two numbers:
The total number of ways to select two numbers from the set {1, 2,...., 11} is given by the combination formula, which is C(11, 2) = 55.
Step 2: Total number of ways to select two even numbers:
Out of the numbers from 1 to 11, there are 6 even numbers (2, 4, 6, 8, 10, 12).
The total number of ways to select two even numbers from these 6 even numbers is given by the combination formula, which is C(6, 2) = 15.
Step 3: Total number of ways to select two odd numbers:
Out of the numbers from 1 to 11, there are 5 odd numbers (1, 3, 5, 7, 9).
The total number of ways to select two odd numbers from these 5 odd numbers is given by the combination formula, which is C(5, 2) = 10.
Step 4: Total number of ways to select one even and one odd number:
Out of the numbers from 1 to 11, there are 6 even numbers (2, 4, 6, 8, 10, 12) and 5 odd numbers (1, 3, 5, 7, 9).
The total number of ways to select one even and one odd number is given by the product of the number of ways to select one even number and the number of ways to select one odd number, which is 6 * 5 = 30.
Step 5: Probability that both numbers are even, given that the sum is even:
The sum of two numbers will be even if both numbers are either even or odd. So, the favorable outcomes for both numbers being even is selecting two even numbers.
The probability that both numbers are even, given that the sum is even, is given by the ratio of the number of ways to select two even numbers to the total number of ways to select two numbers.
Therefore, the required probability is 15/30 = 1/2.
So, the correct answer is option 'B' - 1/2.
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Two integers are selected at random from the set {1, 2,...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :a)2/5b)1/2c)3/5d)7/10Correct answer is option 'A'. Can you explain this answer?
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