2017 AMC 10 B
Complete problem set with solutions and individual problem pages
A radio program has a quiz consisting of multiplechoice questions, each with choices. A contestant wins if he or she gets or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? (2017 AMC 10B Problem, Question#9)
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There are two ways the contestant can win.
Case : The contestant guesses all three right. This can only happen of the time.
Case : The contestant guesses only two right. We pick one of the questions to get wrong, , and this can happen of the time. Thus, .So, in total the two cases combined equals .
Complementary counting is good for solving the problem and checking work if you solved it using the method above.
There are two ways the contestant can lose.
Case : The contestant guesses zero questions correctly.
The probability of guessing incorrectly for each question is . Thus, the probability of guessing all questions incorrectly is .
Case : The contestant guesses one question correctly. There are ways the contestant can guess one question correctly since there are questions. The probability of guessing correctly is .so the probability of guessing one correctly and two incorrectly is .
The sum of the two cases is . This is the complement of what we want to the answer is .
