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If A and B are two independent events and P(AUB) =2/5; P(B) = 1/3. Find P(A).
- a)2/9
- b)-1/3
- c)2/10
- d)1/10
Correct answer is option 'D'. Can you explain this answer?
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If A and B are two independent events and P(AUB) =2/5; P(B) = 1/3. Fin...
To find the probability of event A, we can use the formula of probability of the union of two events:
P(AUB) = P(A) + P(B) - P(A∩B)
Given that P(AUB) = 2/5 and P(B) = 1/3, we can substitute these values into the formula:
2/5 = P(A) + 1/3 - P(A∩B)
Since A and B are independent events, the probability of their intersection, P(A∩B), is equal to the product of their individual probabilities:
P(A∩B) = P(A) * P(B)
Now we can substitute this into the equation:
2/5 = P(A) + 1/3 - P(A) * P(B)
Since we know that P(B) = 1/3, we can substitute this value as well:
2/5 = P(A) + 1/3 - P(A) * (1/3)
Simplifying the equation, we get:
2/5 = P(A) + 1/3 - P(A)/3
Multiplying both sides of the equation by 15 to eliminate the fractions:
6 = 5P(A) + 5/3 - 5P(A)/3
Now, let's simplify further:
6 = 15P(A) + 5 - 5P(A)
Combining like terms:
6 = 10P(A) + 5
Subtracting 5 from both sides:
1 = 10P(A)
Finally, dividing both sides by 10:
P(A) = 1/10
Therefore, the correct answer is option 'D' (1/10).
P(AUB) = P(A) + P(B) - P(A∩B)
Given that P(AUB) = 2/5 and P(B) = 1/3, we can substitute these values into the formula:
2/5 = P(A) + 1/3 - P(A∩B)
Since A and B are independent events, the probability of their intersection, P(A∩B), is equal to the product of their individual probabilities:
P(A∩B) = P(A) * P(B)
Now we can substitute this into the equation:
2/5 = P(A) + 1/3 - P(A) * P(B)
Since we know that P(B) = 1/3, we can substitute this value as well:
2/5 = P(A) + 1/3 - P(A) * (1/3)
Simplifying the equation, we get:
2/5 = P(A) + 1/3 - P(A)/3
Multiplying both sides of the equation by 15 to eliminate the fractions:
6 = 5P(A) + 5/3 - 5P(A)/3
Now, let's simplify further:
6 = 15P(A) + 5 - 5P(A)
Combining like terms:
6 = 10P(A) + 5
Subtracting 5 from both sides:
1 = 10P(A)
Finally, dividing both sides by 10:
P(A) = 1/10
Therefore, the correct answer is option 'D' (1/10).
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