AMC 8 2013 Problem 8
AMC8 2013 年第 8 题
概率★★☆☆☆
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
一枚标准的硬币被掷了 3 次，至少有 2 次连续正面朝上的概率是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
There are ways to flip the coins, in order. There are two ways to get exactly two consecutive heads: HHT and THH. There is only one way to get three consecutive heads: HHH. Therefore, the probability of flipping at least two consecutive heads is .
Let's use complementary counting. To start with, the unfavorable outcomes (in this case, not getting 2 consecutive heads) are: TTT, HTH, and THT. The probability of these three outcomes is , , and , respectively. So the rest is exactly the probability of flipping at least two consecutive heads: . It is the answer .
We can list out all the ways to flip a coin three times: HHH,HHT,HTH,THH,HTT,THT,TTH,TTT. Out of them, only HHH,HHT,THH, have at least two consecutive heads. Since there are three ways to flip at least two consecutive heads, and eight total choices, the answer is .
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