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If P(A) = 0.3, P(B) = 0.2, and P(A and B) = 0.1, determine the following probabilities:

(a) P (A')
(b) P (A U B)
(c) P [(A U B)']
(d) P (A' U B)'

User Mark Maxey
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1 Answer

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Answer:

a)
P(A') = 1-P(A) = 1-0.3=0.7

b)
P(A \cup B) =0.3 +0.2- 0.1=0.4

c)
P[(A \cup B)'] = 1-P(A \cup B) = 1-0.4 =0.6

d) For this case we can find first this probability:


P(A' \cup B) = P(A') +P(B) -P(A' \cap B) = 0.7+0.1 -0.1 =0.7

And then using the complement rule we got:


P[(A' \cup B)']= 1-P(A' \cup B) = 1-0.7=0.3

Explanation:

For this case we define two events A and B with the following probabilities:


P(A) =0.3 , P(B) =0.2, P(A \cap B) =0.1

And we want to find the following probabilities:

(a) P (A')

For this case we can use the complement rule and we got:


P(A') = 1-P(A) = 1-0.3=0.7

(b) P (A U B)

For this case we can use the total probability rule and we got:


P(A \cup B) = P(A) +P(B) -P(A \cap B)

And if we replace we got:


P(A \cup B) =0.3 +0.2- 0.1=0.4

(c) P [(A U B)']

For this case we can use the complement rule again and we have this:


P[(A \cup B)'] = 1-P(A \cup B) = 1-0.4 =0.6

(d) P (A' U B)'

For this case we can find first this probability:


P(A' \cup B) = P(A') +P(B) -P(A' \cap B) = 0.7+0.1 -0.1 =0.7

And then using the complement rule we got:


P[(A' \cup B)']= 1-P(A' \cup B) = 1-0.7=0.3

User CharlesAE
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