Math Problem Statement
In the town of Maplewood a certain type of DVD player is sold at just two stores. 67% of the sales are from store A and the rest of the sales are from store B. 2% of the DVD players sold at store A are defective while 5% of the DVD players sold at store B are defective. If Kate receives one of these DVD players as a gift and finds that it is defective, what is the probability that it came from store A? Express your answer as a percentage rounded to the nearest hundredth without the % sign.
Solution
This is a classic problem in probability using Bayes' Theorem. We are tasked with finding the probability that Kate’s defective DVD player came from Store A, given that it is defective.
Step 1: Define events and probabilities
- Let : The DVD player came from Store A.
- Let : The DVD player came from Store B.
- Let : The DVD player is defective.
From the problem:
- : 67% of DVD players are sold at Store A.
- : 33% of DVD players are sold at Store B.
- : 2% of DVD players from Store A are defective.
- : 5% of DVD players from Store B are defective.
We want to find : the probability that the defective DVD player came from Store A.
Step 2: Bayes' Theorem formula
where is the total probability of a DVD player being defective, computed using the law of total probability:
Step 3: Calculate
Substitute the known values:
Step 4: Calculate
Substitute into Bayes' Theorem:
Convert to a percentage:
Final Answer:
Let me know if you want details or have further questions! Here are some related questions to deepen understanding:
- How does Bayes' Theorem apply in other real-world scenarios?
- What happens to if decreases significantly?
- How does the result change if Store A sells 80% of DVD players instead of 67%?
- Why is it important to consider the total probability in Bayes' Theorem?
- Can you apply the same methodology to non-defective DVD players?
Tip: When solving Bayes' Theorem problems, carefully distinguish between the "given" probabilities and the ones you're solving for!
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Math Problem Analysis
Mathematical Concepts
Probability
Conditional Probability
Bayes' Theorem
Formulas
P(A | D) = (P(D | A) * P(A)) / P(D)
P(D) = P(D | A) * P(A) + P(D | B) * P(B)
Theorems
Bayes' Theorem
Suitable Grade Level
Grades 10-12
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