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Problem of the Week
Problem C and Solution
Fair Game?
Problem
Five identical balls are numbered 1 to 5 and then placed in a bag. The bag is shaken so the
balls are mixed up. A player draws three of the balls from the bag and then determines the
sum of the numbers on the balls. The player wins the game if the sum of the numbers on the
balls is odd. Determine the probability that a player wins the game. Express your answer in
fraction form.
Solution
In order to determine the probability, we must determine the number of ways to obtain a sum
that is odd and divide it by the total number of possible selections of three balls from the bag.
We will count all of the possibilities by systematically listing the possible selections.
Select
Select
Select
Select
Select
Select
1
1
1
2
2
3
and
and
and
and
and
and
2
3
4
3
4
4
and
and
and
and
and
and
one
one
one
one
one
one
higher
higher
higher
higher
higher
higher
number:
number:
number:
number:
number:
number:
123,
134,
145;
234,
245;
345;
124, 125; three possibilities.
135; two possibilities.
one possibility.
235; two possibilities.
one possibility.
one possibility.
By counting the outcomes from each case, there are 3 + 2 + 1 + 2 + 1 + 1 = 10 possible
selections of three balls from the bag. We must now determine how many of these selections
have an odd sum. We could take each of the possibilities, determine the sum and then count
the number which produce an odd sum. However, we will present a different method which
could be useful in other situations. The sum of three numbers is odd in two instances: there
are three odd numbers or there is one odd number and two even numbers.
Selections with three odd numbers: 135.
Selections with one odd number and two even numbers: 124, 234 and 245.
The total number of selections where the sum is odd is 1 + 3 = 4. Therefore the probability of
2
4
= . A game is considered f air if the
winning, by selecting three balls with an odd sum, is
10
5
probability of winning is 50%. In this game, the winning probability, as a percentage, is 40%.
Extending the problem:
If nine balls numbered 1 to 9 are placed in the bag and three balls are removed, then the
10
probability that the sum of the numbers on the balls is odd is
=
˙ 47.6%. Can you verify this?
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