Question

A box contains 11 balls numbered 1, 2, 3, ..., 11. if six balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd

Answer 1

To get an odd number as the sum, you need to take out an odd number of balls wiht an odd number on them.
In total, there are 5 even balls and 6 odd balls.
Possible combinations are
E O , E is even, O is odd
1 5
3 3
5 1

First case: 1 even ball and 5 odd balls
=5C1×6C5
=5×6
=30

Second case: 3 even balls and 3 odd balls
= 5C3×6C3
=10×2
=200

Third case: 5 even balls and 1 odd ball
= 5C5×6C1
=1×6
=6

Total favourable combinations = 30 + 200 + 6 = 236.
Total possible combinations = 11C6 = 462

Probability = 236/462 = 118/231 = 0.5108225... approx or 51.08%