Question

Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. What is the probability that the sum of the two numbers on the balls removed is even?

$\dfrac{4}{9}$

$\dfrac{9}{19}$

$\dfrac{1}{2}$

$\dfrac{10}{19}$

$\dfrac{5}{9}$

Solution:

The sum is even when both balls are odd or both are even. There are $\binom{5}{2} = 10$ all-odd pairs and $\binom{5}{2} = 10$ all-even pairs, for $20$ favorable pairs.

The total number of pairs is $\binom{10}{2} = 45,$ so the probability is $\dfrac{20}{45} = \dfrac49.$

Thus, the correct answer is A .

1987 AMC 8 Problem 25

Solution:

Problem 25 in Other Years