1. **Problem Statement:** You are given probabilities related to customer behavior in a marketing campaign: - Probability a customer opens the email: $P(O) = 0.4$ - Probability a customer makes a purchase after opening the email: $P(P|O) = 0.3$ - Probability a customer does not open the email but still makes a purchase: $P(P|O^c) = 0.05$ We need to find: A) $P(O \cap P)$ B) $P(P)$ C) $P(P^c)$ D) $P(O|P)$ --- 2. **Formulas and Rules:** - Joint probability: $P(O \cap P) = P(O) \times P(P|O)$ - Total probability of purchase: $P(P) = P(O \cap P) + P(O^c \cap P)$ - Complement rule: $P(P^c) = 1 - P(P)$ - Conditional probability: $P(O|P) = \frac{P(O \cap P)}{P(P)}$ --- 3. **Calculations:** **A) Probability customer opens email and makes purchase:** $$ P(O \cap P) = P(O) \times P(P|O) = 0.4 \times 0.3 = 0.12 $$ **B) Probability customer makes a purchase (opens or not):** $$ P(P) = P(O \cap P) + P(O^c \cap P) = 0.12 + (1 - 0.4) \times 0.05 = 0.12 + 0.6 \times 0.05 = 0.12 + 0.03 = 0.15 $$ **C) Probability customer does not make a purchase:** $$ P(P^c) = 1 - P(P) = 1 - 0.15 = 0.85 $$ **D) Probability customer opened email given they made a purchase:** $$ P(O|P) = \frac{P(O \cap P)}{P(P)} = \frac{0.12}{0.15} = 0.8 $$ --- 4. **Interpretation:** - There is a 12% chance a customer opens the email and makes a purchase. - Overall, 15% of customers make a purchase. - 85% of customers do not make a purchase. - Given a purchase was made, there is an 80% chance the customer opened the email. This analysis helps understand customer engagement and effectiveness of the email campaign.