Probability

Question

A bag contains ten identical balls, each numbered with a different number from the integers 1 to 10. Lucky Pix draws three balls from the bag and holds them in her hand. She wins if the smallest numbered ball in her hand is odd and the next smallest numbered ball in her hand is even. Determine the probability that Lucky wins.

Answer 1

There are 120 ways ("hands") to select 3 balls from a set of 10 balls; these are listed in small-to-large order of the selected balls. A “W” indicates that “the smallest is odd” and “the next smallest is even” (thus, a Win).
1  2   3 W
1  2   4 W
1  2   5 W
1  2   6 W
1  2   7 W
1  2   8 W
1  2   9 W
1  2 10 W
1  3   4
1  3   5
1  3   6
1  3   7
1  3   8
1  3   9
1  3 10
1  4   5 W
1  4   6 W
1  4   7 W
1  4   8 W
1  4   9 W
1  4 10 W
1  5   6
1  5   7
1  5   8
1  5   9
1  5 10
1  6   7 W
1  6   8 W
1  6   9 W
1  6 10 W
1  7   8
1  7   9
1  7 10
1  8   9 W
1  8 10 W
1  9 10
2  3   4
2  3   5
2  3   6
2  3   7
2  3   8
2  3   9
2  3 10
2  4   5
2  4   6
2  4   7
2  4   8
2  4   9
2  4 10
2  5   6
2  5   7
2  5   8
2  5   9
2  5 10
2  6   7
2  6   8
2  6   9
2  6 10
2  7   8
2  7   9
2  7 10
2  8   9
2  8 10
2  9 10
3  4   5 W
3  4   6 W
3  4   7 W
3  4   8 W
3  4   9 W
3  4 10 W
3  5   6
3  5   7
3  5   8
3  5   9
3  5 10
3  6   7 W
3  6   8 W
3  6   9 W
3  6 10 W
3  7   8
3  7   9
3  7 10
3  8   9 W
3  8 10 W
3  9 10
4  5   6
4  5   7
4  5   8
4  5   9
4  5 10
4  6   7
4  6   8
4  6   9
4  6 10
4  7   8
4  7   9
4  7 10
4  8   9
4  8 10
4  9 10
5  6   7 W
5  6   8 W
5  6   9 W
5  6 10 W
5  7   8
5  7   9
5  7 10
5  8   9 W
5  8 10 W
5  9 10
6  7   8
6  7   9
6  7 10
6  8   9
6  8 10
6  9 10
7  8   9 W
7  8 10 W
7  9 10
8  9 10

40 of the possible 120 combinations result in a Win. This is a probability of 40/120 = 1/3.

Answer 2

₁₀C₃ = 120              total possibilities

 

Winners:

2, 3

4, 5

6, 7

8, 9

 

4/120

1/30