When the sum of three numbers is odd, the three numbers should be

1) three odd

2) one odd, two even

Probability of picking three odd numbers

= 1/2 * 1/2 * 1/2

= 1/8

Probability of picking one odd number and two even numbers

= 1/2 * 1/2 * 1/2 * 3

= 3/8

(multiplied by 3 since there are 3 combinations - ODD, DOD, DDO)

Thus, 1/8 + 3/8 = 1/2

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Or, from the fact that the probability of picking an odd number and the probability of picking an even number is the same, I guess we can do the following also.

Three numbers can be as below;

1. odd, odd, odd

2. odd, odd, even

3. odd, even, odd

4. even, odd, odd

5. odd, even, even

6. even, odd, even

7. even, even, odd

8. even, even, even

There are 8 cases in total and 4 cases(1, 5, 6, 7) will yield an odd number.

Thus it's 4/8 = 1/2.

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Or..what about this approach?

The probability of picking an odd or an even number is the same.

And the result of summing three numbers will always yield odd or even number. This or that. No others. So 1/2.

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