AMC 8 1987 Problem 25
AMC8 1987 年第 25 题
概率★★★☆☆
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. The probability that the sum of the two numbers on the balls removed is even is
一个罐子中有编号 1 到 10 的十个球。Jack 从罐中随机取出一个球。然后 Jill 从罐中随机取一个不同的球。取出的两个球上的数字之和为偶数的概率是
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
At first, there are 5 even balls and 5 odd balls. Therefore, after Jack chooses a ball, the jar will have 4 balls with the same parity (odd/even) as Jack's ball and 5 balls with the opposite parity. For the sum to be even, Jill must choose a ball with the same parity as Jack's ball. The probability that this happens is .
Jack has 10 choices for his ball, and after he makes his selection Jill will have 9 choices for hers. Therefore, there are combinations of selections, and each combination is equally likely.
For the sum of their numbers to be even, Jack and Jill must either both select odd numbers or both select even numbers.
If they both select even numbers, Jack has 5 choices for his ball, and Jill will have 4 choices for hers, for a total of combinations of selections.
If they both select odd numbers, Jack has 5 choices for his ball, and Jill will have 4 choices for hers, for a total of combinations of selections.
Therefore, of the 90 combinations of selections, 20+20=40 yield an even sum. Therefore, the probability that the sum is even is .
- Complementary Counting
An alternative approach would be to calculate the probability the sum of the two numbers is odd, and subtract that from the total. We see that there is only one way to get an odd sum, where the two numbers must be of opposite parity.
Here, we have two cases the two numbers are, respectively, (even, odd), and (odd, even). The probability to get an even number as the first number is , and an odd number as the second number gives us a probability of . Multiplying the two, we get , and because both cases have equal probability, we have the probability for the sum to be odd is .
Finally, subtracting this from 1 gives us .
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