If P(A ∪ B) = 5/9, P(A ∪ B̅) = 13/27, P(A) = 11/18, then the odds against the event B are:
7/2
Given:
P(A ∪ B) = 5/9
P(A ∪ B̅) = 13/27
P(A) = 11/18
Step 1: Use the identity
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
5/9 = 11/18 + P(B) − P(A ∩ B)
Multiplying both sides by 18:
10 = 11 + 18P(B) − 18P(A ∩ B)
18P(B) − 18P(A ∩ B) = −1 ...(1)
Step 2: Use
P(A ∪ B̅) = P(A) + P(B̅) − P(A ∩ B̅)
13/27 = 11/18 + (1 − P(B)) − (11/18 − P(A ∩ B))
13/27 = 1 − P(B) + P(A ∩ B)
Multiplying both sides by 27:
13 = 27 − 27P(B) + 27P(A ∩ B)
27P(B) − 27P(A ∩ B) = 14 ...(2)
Step 3: Solving equations (1) and (2)
From the equations, we obtain:
P(B) = 2/9
Step 4: Find odds against event B
Odds against B = P(B̅) : P(B)
= (1 − 2/9) : 2/9
= 7/9 : 2/9
= 7 : 2
Final Answer:
Odds against the event B = 7 : 2
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