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 AA: The DVD player came from Store A.
  • Let BB: The DVD player came from Store B.
  • Let DD: The DVD player is defective.

From the problem:

  • P(A)=0.67P(A) = 0.67: 67% of DVD players are sold at Store A.
  • P(B)=1P(A)=0.33P(B) = 1 - P(A) = 0.33: 33% of DVD players are sold at Store B.
  • P(DA)=0.02P(D \mid A) = 0.02: 2% of DVD players from Store A are defective.
  • P(DB)=0.05P(D \mid B) = 0.05: 5% of DVD players from Store B are defective.

We want to find P(AD)P(A \mid D): the probability that the defective DVD player came from Store A.

Step 2: Bayes' Theorem formula

P(AD)=P(DA)P(A)P(D)P(A \mid D) = \frac{P(D \mid A) P(A)}{P(D)} where P(D)P(D) is the total probability of a DVD player being defective, computed using the law of total probability: P(D)=P(DA)P(A)+P(DB)P(B)P(D) = P(D \mid A) P(A) + P(D \mid B) P(B)

Step 3: Calculate P(D)P(D)

Substitute the known values: P(D)=(0.02)(0.67)+(0.05)(0.33)P(D) = (0.02)(0.67) + (0.05)(0.33) P(D)=0.0134+0.0165=0.0299P(D) = 0.0134 + 0.0165 = 0.0299

Step 4: Calculate P(AD)P(A \mid D)

Substitute into Bayes' Theorem: P(AD)=(0.02)(0.67)0.0299P(A \mid D) = \frac{(0.02)(0.67)}{0.0299} P(AD)=0.01340.02990.4482P(A \mid D) = \frac{0.0134}{0.0299} \approx 0.4482

Convert to a percentage: P(AD)=44.82P(A \mid D) = 44.82

Final Answer:

44.82\boxed{44.82}

Let me know if you want details or have further questions! Here are some related questions to deepen understanding:

  1. How does Bayes' Theorem apply in other real-world scenarios?
  2. What happens to P(AD)P(A \mid D) if P(DB)P(D \mid B) decreases significantly?
  3. How does the result change if Store A sells 80% of DVD players instead of 67%?
  4. Why is it important to consider the total probability P(D)P(D) in Bayes' Theorem?
  5. 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