4. Probability
Multiplication Rule: Independent Events
3:18 minutesProblem 3.2.33
Textbook Question
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).
33. P(A) = 2/3, P(A') = 1/3, P(B|A) = 1/5 , and P(B|A') = 1/2
Verified step by step guidance1
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 us calculate the conditional probability of event A given that event B has occurred.
Step 2: Substitute the given values into the formula. From the problem, P(A) = 2/3, P(A') = 1/3, P(B|A) = 1/5, and P(B|A') = 1/2. The formula becomes: P(A|B) = ((2/3) * (1/5)) / ((2/3) * (1/5) + (1/3) * (1/2)).
Step 3: Simplify the numerator. Multiply P(A) and P(B|A): (2/3) * (1/5).
Step 4: Simplify the denominator. First, calculate (P(A) * P(B|A)) and (P(A') * P(B|A')). Then, add these two results together: ((2/3) * (1/5)) + ((1/3) * (1/2)).
Step 5: Divide the simplified numerator by the simplified denominator to find P(A|B). This will give you the conditional probability of A given B.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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(A) * P(B|A) + P(A') * P(B|A')). This theorem is particularly useful in scenarios where prior knowledge about the events is available.
Conditional Probability
Conditional probability is the measure of 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 event A occurring under the condition that event 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.
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Prior and Posterior Probabilities
In the context of Bayes' Theorem, prior probability refers to the initial assessment of the likelihood of an event before new evidence is considered, denoted as P(A). Posterior probability, on the other hand, is the updated probability of the event after taking into account the new evidence, represented as P(A|B). Distinguishing between these two types of probabilities is essential for correctly applying Bayes' Theorem to update beliefs based on new data.
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