Question

P(A) = 0.42, P(B|A) = 0.66, and P(B|A') = 0.25. Find the following probabilities:

a) P(A)
b) P(B|A)
c) P(B|A')
d) P(B)
e) P(B')
f) P(A|B)
g) P(A|B')
h) P(A'|B)
i) P(A'|B')'

Answer 1

This problem involves calculating various probabilities, including conditional probabilities and complements, using the given information and applying the rules of conditional probability and the Law of Total Probability.

Step-by-step explanation:

In order to solve this probability question, we can use the rules of conditional probability and the Law of Total Probability. Given are the following probabilities: P(A) = 0.42, P(B|A) = 0.66, and P(B|A') = 0.25. Let's find the required probabilities one by one.

• a. P(A) is given as 0.42.
• b. P(B|A) is given as 0.66.
• c. P(B|A') is given as 0.25.
• d. P(B) can be found using the Law of Total Probability:
  P(B) = P(B|A)P(A) + P(B|A')P(A') = (0.66)(0.42) + (0.25)(1 - 0.42) = 0.2762 + 0.145 = 0.4212.
• e. P(B') is the complement of P(B):
  P(B') = 1 - P(B) = 1 - 0.4212 = 0.5788.
• f. P(A|B) can be found using Bayes' theorem:
  P(A|B) = (P(B|A)P(A)) / P(B) = (0.66)(0.42) / 0.4212 ≈ 0.6578.
• g. P(A|B') is also found using Bayes' theorem or conditional probability:
  P(A|B') can be calculated using the information provided and the complement rules.
• h. P(A'|B) represents the probability of A' given B:
  P(A'|B) = 1 - P(A|B) ≈ 0.3422.
• i. P(A'|B') represents the probability of A' given B':
  P(A'|B') = 1 - P(A|B') can be calculated using the information provided and the complement rules.