Item Availability Probability

Solution & Explanation

Understanding the Problem: We are given two warehouses, A and B, each with a certain probability of having a particular item X in stock. The goal is to find the probability that item X is listed on Amazon's website, which occurs if the item is available in at least one of the warehouses.

Given Probabilities:

• Probability that item X is available at Warehouse A, $P(A) = 0.6$
• Probability that item X is available at Warehouse B, $P(B) = 0.8$

Objective: We need to calculate the probability that item X is available at either Warehouse A or Warehouse B (or both), denoted as $P(A \cup B)$.

Probability Theory Used: The probability of the union of two events A and B is given by: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ where:

• $P(A \cap B)$ is the probability that item X is available at both Warehouse A and Warehouse B.

Assumption: Assume that the availability of item X at Warehouse A and Warehouse B are independent events. This allows us to calculate $P(A \cap B)$ as: $P(A \cap B) = P(A) \times P(B)$

Calculation:

1. Calculate $P(A \cap B)$: $P(A \cap B) = 0.6 \times 0.8 = 0.48$

2. Calculate $P(A \cup B)$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $P(A \cup B) = 0.6 + 0.8 - 0.48 = 0.92$

Conclusion: The probability that item X will be listed on Amazon's website, meaning it is available at either Warehouse A or Warehouse B, is 0.92 or 92%.

Alternative Approach Using Complement Rule: Another way to find $P(A \cup B)$ is by using the complement rule:

• Calculate the probability that item X is not available at both warehouses, $P(A' \cap B')$: $P(A' \cap B') = (1 - P(A)) \times (1 - P(B)) = 0.4 \times 0.2 = 0.08$
• The probability that item X is available at either warehouse is: $P(A \cup B) = 1 - P(A' \cap B') = 1 - 0.08 = 0.92$

This alternative approach confirms the result obtained using the union probability formula. Thus, the probability that item X is listed on Amazon's website is indeed 92%.