The events A and B are such that P(A\B)=0.4, P(B\A)=0.25 and P(A n B)=0.12 compute the following;
1. P(A) and p(B)
2. P(A n B') and P(A' n B)
3.P(A u B)and P[(A u B)']
To compute the requested probabilities, we will use conditional probability and set theory principles.
1. P(A) and P(B):
To find P(A) and P(B), we need to consider the information given and use the formula for conditional probability.
P(A|B) = P(A ∩ B) / P(B)
Given: P(A|B) = 0.4 and P(A ∩ B) = 0.12
We can rearrange the formula to get:
P(A ∩ B) = P(B) * P(A|B)
P(B) = P(A ∩ B) / P(A|B)
P(B) = 0.12 / 0.4
P(B) = 0.3
Similarly, we can find the value of P(A):
P(B|A) = P(A ∩ B) / P(A)
Given: P(B|A) = 0.25 and P(A ∩ B) = 0.12
P(A) = P(A ∩ B) / P(B|A)
P(A) = 0.12 / 0.25
P(A) = 0.48
Therefore, P(A) = 0.48 and P(B) = 0.3.
2. P(A ∩ B') and P(A' ∩ B):
To calculate these probabilities, we need the complement of each event.
P(A ∩ B') = P(A) - P(A ∩ B)
P(A' ∩ B) = P(B) - P(A ∩ B)
We know the values of P(A) = 0.48, P(B) = 0.3, and P(A ∩ B) = 0.12:
P(A ∩ B') = 0.48 - 0.12
P(A ∩ B') = 0.36
P(A' ∩ B) = 0.3 - 0.12
P(A' ∩ B) = 0.18
Therefore, P(A ∩ B') = 0.36 and P(A' ∩ B) = 0.18.
3. P(A ∪ B) and P[(A ∪ B)']:
To calculate the union of two events (A ∪ B), we need to use the formula:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Given the values of P(A) = 0.48, P(B) = 0.3, and P(A ∩ B) = 0.12:
P(A ∪ B) = 0.48 + 0.3 - 0.12
P(A ∪ B) = 0.66
To find the complement of (A ∪ B), we use the formula:
P[(A ∪ B)'] = 1 - P(A ∪ B)
P[(A ∪ B)'] = 1 - 0.66
P[(A ∪ B)'] = 0.34
Therefore, P(A ∪ B) = 0.66 and P[(A ∪ B)'] = 0.34.