Math Problem Statement
A multiple choice exam has 10 questions. Each question has 3 possible answers, of which one is correct. A student knows the correct answers to 4 questions and guesses the answers to the other 6 questions.
It turns out that the student answered the first question correctly. What are the chances that the student was merely guessing?
1 point
(13)(610)+(1)(410)(13)(610)(31)(106)(31)(106)+(1)(104)
(13)(410)+(1)(610)(13)(410)(31)(104)(31)(104)+(1)(106)
(13)(410)(13)(410)+(1)(610)(31)(104)+(1)(106)(31)(104)
(13)(610)(13)(610)+(1)(410)(31)(106)+(1)(104)(31)(106)
Solution
To solve this problem, we need to use Bayes' theorem to find the probability that the student was guessing given that they answered the first question correctly.
Let's denote:
- as the event that the student was guessing on the first question.
- as the event that the student knew the answer to the first question.
- as the event that the student answered the first question correctly.
We need to find , the probability that the student was guessing given that they answered the first question correctly.
By Bayes' theorem:
First, let's calculate each component:
-
: The probability that the student was guessing on the first question. Since the student knows the answers to 4 out of 10 questions and guesses on the remaining 6:
-
: The probability that the student knew the answer to the first question.
-
: The probability of answering correctly given that the student was guessing. Since there is only one correct answer out of three possible answers:
-
: The probability of answering correctly given that the student knew the answer.
-
: The total probability of answering the first question correctly. This is calculated by considering both possibilities (knowing the answer and guessing):
Now, applying Bayes' theorem:
Thus, the probability that the student was merely guessing given that they answered the first question correctly is .
Do you want more details or have any questions?
Here are 5 more questions you might find interesting:
- How do you apply Bayes' theorem to different types of probability problems?
- Can you explain the concept of prior and posterior probabilities?
- How does conditional probability differ from joint probability?
- What are some common real-life applications of Bayes' theorem?
- How can you calculate the probability of dependent events in a sequence?
Tip: When dealing with probabilities, always carefully consider all possible scenarios and how they contribute to the total probability.
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Math Problem Analysis
Mathematical Concepts
Probability
Bayes' Theorem
Formulas
-
Theorems
Bayes' Theorem
Suitable Grade Level
Advanced High School
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