To solve this problem, we need to use Bayes' Theorem, which allows us to find the probability of an event given that another event has already occurred. The theorem is expressed as:

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Where:

• $P(A|B)$ is the probability that the purchaser has the basic model given that they have an extended warranty.
• $P(B|A)$ is the probability that a purchaser with the basic model has an extended warranty.
• $P(A)$ is the probability of a purchaser having the basic model.
• $P(B)$ is the total probability that a purchaser has an extended warranty.

Let's define the events:

• $A_1$: Purchaser buys the basic model.
• $A_2$: Purchaser buys the deluxe model.
• $B$: Purchaser has an extended warranty.

We are asked to find $P(A_1 | B)$, the probability that a purchaser bought the basic model given that they have an extended warranty.

Step 1: Gather the probabilities

From the problem:

• $P(A_1) = 0.33$ (33% of purchasers buy the basic model).
• $P(A_2) = 1 - P(A_1) = 0.67$ (67% of purchasers buy the deluxe model).
• $P(B|A_1) = 0.41$ (41% of basic model buyers purchase an extended warranty).
• $P(B|A_2) = 0.46$ (46% of deluxe model buyers purchase an extended warranty).

Step 2: Calculate the total probability of having an extended warranty, $P(B)$

We use the law of total probability to find $P(B)$:

$P(B) = P(B|A_1) \cdot P(A_1) + P(B|A_2) \cdot P(A_2)$

Substituting the values:

$P(B) = (0.41 \cdot 0.33) + (0.46 \cdot 0.67)$

Step 3: Apply Bayes' Theorem

Now we can apply Bayes' Theorem to find $P(A_1 | B)$:

$P(A_1 | B) = \frac{P(B|A_1) \cdot P(A_1)}{P(B)}$

Substitute the values and compute the result.

Let's calculate this step-by-step.The probability that a randomly selected purchaser with an extended warranty has bought the basic model is approximately 0.3051 (or 30.51%).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the probability change if the percentage of deluxe buyers purchasing a warranty increased?
2. What happens to the overall result if more people start buying the basic model instead of the deluxe model?
3. Can Bayes' Theorem be applied if there are more than two types of camera models?
4. What is the role of the law of total probability in this problem?
5. How does the result relate to real-world marketing strategies for basic and deluxe products?

Tip: Bayes' Theorem is widely used in medical diagnostics, spam filtering, and machine learning to make informed predictions based on prior knowledge.