Let A,B be events in a sample space. you are given the following probabilities:
P(A)=0.8, P(B|A)=0.3, P(B|A^c)=0.8
Find:
1) P(B)
2) P(A|B)
3) P(A^c|B^c)
In the following, A' will be used in place of Ac for ease of typography.
(1) since A and A' are partitions of A, we can apply the law of total probability
P(B)=P(B|A)*P(A)+P(B|A')*P(A')
(2) Use Bayes Theorem
P(A|B)
=P(A∩B)/P(B) ... def of cond. prob
=P(B∩A)/P(B) ... commutativity
=[P(B|A)*P(A)]/P(B) ... def of cond. prob
(3) We need to use
P(A'∩B')=1-P(A∪B) ... complement
and
P(A∪B)=P(A)+P(B)-P(A∩B) ... prob. of union of sets
P(A'|B')
=P(A'∩B')/P(B') .. def of cond.prob
=(1-P(A∪B))/(1-P(B)) ... compl
=(1-P(B∪A))/(1-P(B)) ... commutativity
=[1-(P(A)+P(B)-P(A∩B)]/(1-P(B)) ... union of sets
=[1-(P(A)+P(B)-P(B∩A)]/(1-P(B)) ... commutativity
=[1-(P(A)+P(B)-P(B|A)*P(A)]/(1-P(B)) ... def. of cond. prob.
Please check and understand every line. Post if you have questions.
To find the probabilities in this scenario, we can use conditional probability, complement rule, and the formula for joint events.
1) P(B):
We can use the law of total probability to find P(B). The law of total probability states that for any event B, we can express its probability as the sum of the probabilities of B given A and B given A complement:
P(B) = P(B|A) * P(A) + P(B|A^c) * P(A^c)
Given:
P(A) = 0.8
P(B|A) = 0.3
P(B|A^c) = 0.8
Substituting these values into the formula:
P(B) = 0.3 * 0.8 + 0.8 * (1 - 0.8)
Simplifying the expression:
P(B) = 0.24 + 0.16
P(B) = 0.4
Therefore, P(B) is equal to 0.4.
2) P(A|B):
We can use Bayes' theorem to find P(A|B). Bayes' theorem states that for any events A and B:
P(A|B) = (P(B|A) * P(A)) / P(B)
Given:
P(A) = 0.8
P(B|A) = 0.3
P(B) = 0.4
Substituting these values into the formula:
P(A|B) = (0.3 * 0.8) / 0.4
Simplifying the expression:
P(A|B) = 0.24 / 0.4
P(A|B) = 0.6
Therefore, P(A|B) is equal to 0.6.
3) P(A^c|B^c):
We can apply the complement rule to find P(A^c|B^c). The complement rule states that for any event A:
P(A) + P(A^c) = 1
Given:
P(A) = 0.8
Rearranging the equation:
P(A^c) = 1 - P(A)
Substituting the given value:
P(A^c) = 1 - 0.8
P(A^c) = 0.2
Since A^c and B^c are complements, we can write P(A^c|B^c) as P(B^c|A^c) using Bayes' theorem:
P(A^c|B^c) = (P(B^c|A^c) * P(A^c)) / P(B^c)
Given:
P(B^c|A^c) = 1 - P(B|A^c) = 1 - 0.8 = 0.2
P(A^c) = 0.2
Substituting these values into the formula:
P(A^c|B^c) = (0.2 * 0.2) / P(B^c)
To find P(B^c), we can use the complement rule similarly to how we found P(A^c). Since P(B) + P(B^c) = 1, we have:
P(B^c) = 1 - P(B)
Given:
P(B) = 0.4
Substituting this value into the formula:
P(A^c|B^c) = (0.2 * 0.2) / (1 - 0.4)
Simplifying the expression:
P(A^c|B^c) = 0.04 / 0.6
P(A^c|B^c) = 0.067
Therefore, P(A^c|B^c) is approximately equal to 0.067.