AMC 8 2011 Problem 18
AMC8 2011 年第 18 题
概率★★★☆☆
A fair 6 sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?
一个标准的 6 面骰子掷了两次，那么第一次正面朝上的数大于或等于第二次正面朝上的数的概率是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
There are ways to roll the two dice, and 6 of them result in two of the same number. Out of the remaining 36-6=30 ways, the number of rolls where the first die is greater than the second should be the same as the number of rolls where the second die is greater than the first. In other words, there are ways the first roll can be greater than the second. The probability the first number is greater than or equal to the second number is
If we list out some ways the first dice is greater than the second dice, we might see a pattern. If dice 2 is 1, then dice 1 can be 1, 2, 3, 4, 5, or 6. If dice 2 is 2, then dice 1 can be 2, 3, 4, 5, or 6. If dice 2 is 3, then dice 1 can be 3, 4, 5, or 6. In the first case, dice 1 had six values it could land on. In the second case, dice 1 had five values it could land on. In the third case, dice 1 had four values it could land on. If we keep going, we would eventually get the pattern 6+5+4+3+2+1=21 as the number of total values dice 1 could land on if it's greater or equal to dice 2. Since there are 36 ways two dice could land, our answer is , or .
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