"According to Bayes’ Theorem, the probability of event A , given that event B has occurred, is
P(A|B) = P(A) * P(B|A)P(A) * P(B|A) + P(A') * P(B|A').
In Exercises 33–38, use Bayes’ Theorem to find P(A|B).
34. P(A) = 3/8, P(A') = 5/8, P(B|A) = 2/3 , and P(B|A') = 3/5 "

Verified step by step guidance

1. Step 1: Recall Bayes' Theorem formula: P(A|B) = (P(A) * P(B|A)) / (P(A) * P(B|A) + P(A') * P(B|A')). This formula helps calculate the conditional probability of event A given that event B has occurred.
2. Step 2: Substitute the given values into the formula. From the problem, P(A) = 3/8, P(A') = 5/8, P(B|A) = 2/3, and P(B|A') = 3/5. Replace these values in the formula.
3. Step 3: Calculate the numerator of the formula, which is P(A) * P(B|A). Multiply 3/8 by 2/3.
4. Step 4: Calculate the denominator of the formula, which is P(A) * P(B|A) + P(A') * P(B|A'). First, calculate P(A') * P(B|A') by multiplying 5/8 by 3/5. Then, add this result to the numerator calculated in Step 3.
5. Step 5: Divide the numerator (from Step 3) by the denominator (from Step 4) to find P(A|B). This will give you the conditional probability of event A given that event B has occurred.

Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental principle in probability theory that describes how to update the probability of a hypothesis based on new evidence. It states that the probability of event A given event B, denoted as P(A|B), can be calculated using the formula P(A|B) = P(A) * P(B|A) / P(B). This theorem is particularly useful in scenarios where we want to revise our beliefs in light of new data.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which represents the probability of A occurring under the condition that B is true. Understanding conditional probability is crucial for applying Bayes' Theorem, as it allows us to assess how the occurrence of one event influences the likelihood of another.

Prior and Posterior Probabilities

In the context of Bayes' Theorem, prior probability refers to the initial assessment of the likelihood of an event before considering new evidence, denoted as P(A). Posterior probability, on the other hand, is the updated probability after taking into account the new evidence, represented as P(A|B). Distinguishing between these two types of probabilities is essential for understanding how new information can alter our beliefs about uncertain events.