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A box contains 50 balls, numbered from 1 to 50. If three balls are drawn at random with replacement. What is the probability that sum of the numbers are odd?
- a)1/2
- b)1/3
- c)2/7
- d)1/5
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A box contains 50 balls, numbered from 1 to 50. If three balls are dra...
There are 25 odd and 25 even numbers from 1 to 50.
Sum will be odd if = odd + odd + odd, odd + even + even, even + odd + even, even+ even + odd
P = (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)
=4/8 = ½
Sum will be odd if = odd + odd + odd, odd + even + even, even + odd + even, even+ even + odd
P = (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)
=4/8 = ½
Most Upvoted Answer
A box contains 50 balls, numbered from 1 to 50. If three balls are dra...
Solution:
To find the probability that the sum of numbers drawn is odd, we need to consider the following cases:
Case 1: The sum of three odd numbers is odd.
In this case, we need to select three odd numbers from the 25 odd numbers in the box. The probability of selecting an odd number is 1/2 (since there are 25 odd and 25 even numbers in the box). Therefore, the probability of selecting three odd numbers is (1/2)^3 = 1/8. The sum of three odd numbers is always odd.
Case 2: The sum of two even numbers and one odd number is odd.
In this case, we need to select two even numbers and one odd number from the box. The probability of selecting an even number is also 1/2. Therefore, the probability of selecting two even numbers is (1/2)^2 = 1/4. The probability of selecting one odd number is 1/2. Therefore, the probability of selecting two even numbers and one odd number is (1/4) x (1/2) = 1/8. The sum of two even numbers is always even. Therefore, the sum of two even numbers and one odd number is odd.
Case 3: The sum of one even number and two odd numbers is odd.
In this case, we need to select one even number and two odd numbers from the box. The probability of selecting an even number is 1/2. The probability of selecting an odd number is also 1/2. Therefore, the probability of selecting one even number and two odd numbers is (1/2) x (1/2)^2 = 1/8. The sum of one even number and two odd numbers is always odd.
Therefore, the total probability of getting an odd sum is the sum of probabilities of the above three cases, which is:
1/8 + 1/8 + 1/8 = 3/8.
But this is the probability of getting an odd sum in one draw. Since we are drawing three balls with replacement, we need to consider all the possible combinations of three balls. The total number of possible combinations of three balls is:
50 x 50 x 50 = 125,000.
Therefore, the probability of getting an odd sum in three draws is:
(3/8)^3 = 27/512.
Therefore, the correct answer is option A) 1/2.
To find the probability that the sum of numbers drawn is odd, we need to consider the following cases:
Case 1: The sum of three odd numbers is odd.
In this case, we need to select three odd numbers from the 25 odd numbers in the box. The probability of selecting an odd number is 1/2 (since there are 25 odd and 25 even numbers in the box). Therefore, the probability of selecting three odd numbers is (1/2)^3 = 1/8. The sum of three odd numbers is always odd.
Case 2: The sum of two even numbers and one odd number is odd.
In this case, we need to select two even numbers and one odd number from the box. The probability of selecting an even number is also 1/2. Therefore, the probability of selecting two even numbers is (1/2)^2 = 1/4. The probability of selecting one odd number is 1/2. Therefore, the probability of selecting two even numbers and one odd number is (1/4) x (1/2) = 1/8. The sum of two even numbers is always even. Therefore, the sum of two even numbers and one odd number is odd.
Case 3: The sum of one even number and two odd numbers is odd.
In this case, we need to select one even number and two odd numbers from the box. The probability of selecting an even number is 1/2. The probability of selecting an odd number is also 1/2. Therefore, the probability of selecting one even number and two odd numbers is (1/2) x (1/2)^2 = 1/8. The sum of one even number and two odd numbers is always odd.
Therefore, the total probability of getting an odd sum is the sum of probabilities of the above three cases, which is:
1/8 + 1/8 + 1/8 = 3/8.
But this is the probability of getting an odd sum in one draw. Since we are drawing three balls with replacement, we need to consider all the possible combinations of three balls. The total number of possible combinations of three balls is:
50 x 50 x 50 = 125,000.
Therefore, the probability of getting an odd sum in three draws is:
(3/8)^3 = 27/512.
Therefore, the correct answer is option A) 1/2.
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