Financial Risk Manager Part 1
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Let:
- A be the event that a customer chooses/orders apple juice
- M be the event that a customer chooses mango juice
- S be the event that a customer chooses passion fruit
We can easily establish that:
- 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
We need to determine P(A ∩ M ∩ S):
Explanation:
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.