Answer: 0.10

## Explanation To find P(A ∩ M ∩ S), we use the addition rule for three sets: **Formula:** P(A ∪ M ∪ S) = P(A) + P(M) + P(S) − P(M ∩ A) − P(M ∩ S) − P(A ∩ S) + P(A ∩ M ∩ S) **Given:** - P(S) = 0.45 - P(M ∩ A) = 0.19 - P(M ∩ S) = 0.15 - P(A ∩ S) = 0.25 - P(M ∪ S) = 0.6 - P(A ∪ S) = 0.84 - P(A ∪ M ∪ S) = 0.9 **Step 1: Find P(M)** Using P(M ∪ S) = P(M) + P(S) − P(M ∩ S): 0.6 = P(M) + 0.45 − 0.15 P(M) = 0.6 + 0.15 − 0.45 = 0.3 **Step 2: Find P(A)** Using P(A ∪ S) = P(A) + P(S) − P(A ∩ S): 0.84 = P(A) + 0.45 − 0.25 P(A) = 0.84 − 0.45 + 0.25 = 0.64 **Step 3: Apply the three-set addition rule** 0.9 = 0.64 + 0.3 + 0.45 − 0.19 − 0.15 − 0.25 + P(A ∩ M ∩ S) 0.9 = 0.8 + P(A ∩ M ∩ S) P(A ∩ M ∩ S) = 0.9 − 0.8 = 0.10 Therefore, the probability that a customer chooses all three juices (apple, mango, and passion fruit) is **0.10**.

Author: Tanishq Prabhu