The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3 - Leaving Cert Mathematics - Question 1 - 2012
Question 1
The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3.
(a) Find P(A ∪ B)
(b) Find P(A | B)
(c) State whether A and B are independent events, ... show full transcript
Worked Solution & Example Answer:The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3 - Leaving Cert Mathematics - Question 1 - 2012
Step 1
Find P(A ∪ B)
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Answer
To find the probability of the union of two events A and B, we can use the formula:
P(A∪B)=P(A)+P(B)−P(A∩B)
Substituting in the given values:
P(A∪B)=0.7+0.5−0.3 =0.9
Thus, the probability of A union B is 0.9.
Step 2
Find P(A | B)
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Answer
The conditional probability of A given B is found using the formula:
P(A∣B)=P(B)P(A∩B)
Substituting in the given values:
P(A∣B)=0.50.3=0.6
Therefore, the conditional probability P(A | B) is 0.6.
Step 3
State whether A and B are independent events, and justify your answer.
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Answer
To determine if A and B are independent, we check if:
P(A∩B)=P(A)P(B)
If A and B were independent, then:
P(A)P(B)=0.7×0.5=0.35
However, we know:
P(A∩B)=0.3
Since 0.3 is not equal to 0.35, we conclude that A and B are NOT independent events.
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