Financial Risk Manager Part 1
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A fruit juice shop allows customers to choose apple juice, mango juice or passion juice. The probability of a customer ordering passion juice is 0.45, mango juice and apple juice 0.19, passion juice and mango juice 0.15, passion juice and apple juice 0.25, passion juice or mango juice 0.6, passion juice or apple juice 0.84, and 0.9 for at least one of them.
Find the probability that a customer orders all the three juices.
Other
Community
NNikitesh
A
0.64
B
0.1
C
0.3
D
0.25
Explanation:
Solution:
Let:
- A = event that a customer orders apple juice
- M = event that a customer orders mango juice
- S = event that a customer orders passion juice
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
We need to find P(A ∩ M ∩ S).
Step 1: Find P(M) Using the formula for union of two events: P(M ∪ S) = P(M) + P(S) - P(M ∩ S) 0.6 = P(M) + 0.45 - 0.15 0.6 = P(M) + 0.3 P(M) = 0.6 - 0.3 = 0.3
Step 2: Find P(A) Using the formula for union of two events: P(A ∪ S) = P(A) + P(S) - P(A ∩ S) 0.84 = P(A) + 0.45 - 0.25 0.84 = P(A) + 0.2 P(A) = 0.84 - 0.2 = 0.64
Step 3: Apply the inclusion-exclusion principle for three events P(A ∪ M ∪ S) = P(A) + P(M) + P(S) - P(A ∩ M) - P(A ∩ S) - P(M ∩ S) + P(A ∩ M ∩ S) 0.9 = 0.64 + 0.3 + 0.45 - 0.19 - 0.25 - 0.15 + P(A ∩ M ∩ S) 0.9 = 1.39 - 0.59 + P(A ∩ M ∩ S) 0.9 = 0.8 + P(A ∩ M ∩ S) P(A ∩ M ∩ S) = 0.9 - 0.8 = 0.1
Therefore, the probability that a customer orders all three juices is 0.1.
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