4. Probability
Basic Concepts of Probability
2:45 minutesProblem 4.2.2
Textbook Question
Notation When randomly selecting adults, let M denote the event of randomly selecting a male and let B denote the event of randomly selecting someone with blue eyes. What does P (M|B) represent? Is P (M|B) the same as P (B|M)?
Verified step by step guidance1
Step 1: Understand the notation P(M|B). This represents the conditional probability of event M (selecting a male) given that event B (selecting someone with blue eyes) has already occurred. Mathematically, it is expressed as P(M|B) = P(M ∩ B) / P(B), where P(M ∩ B) is the probability of both events M and B occurring, and P(B) is the probability of event B.
Step 2: Understand the notation P(B|M). This represents the conditional probability of event B (selecting someone with blue eyes) given that event M (selecting a male) has already occurred. Mathematically, it is expressed as P(B|M) = P(M ∩ B) / P(M), where P(M ∩ B) is the probability of both events M and B occurring, and P(M) is the probability of event M.
Step 3: Compare P(M|B) and P(B|M). These two probabilities are not necessarily the same because they depend on different conditions. P(M|B) is conditioned on the event of selecting someone with blue eyes, while P(B|M) is conditioned on the event of selecting a male. The denominators in their formulas, P(B) and P(M), are generally different unless the probabilities of these events are equal.
Step 4: Recognize the importance of the joint probability P(M ∩ B). Both P(M|B) and P(B|M) share the same numerator, which is the joint probability of selecting a male with blue eyes. However, the difference lies in the denominator, which changes based on the given condition.
Step 5: Conclude that P(M|B) and P(B|M) are not the same in general. They represent different conditional probabilities and depend on the specific probabilities of the events M and B. To determine their values, you would need the probabilities P(M), P(B), and P(M ∩ B).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conditional Probability
Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as 'the probability of A given B.' In this context, P(M|B) represents the probability of selecting a male given that the individual has blue eyes.
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Bayes' Theorem
Bayes' Theorem is a fundamental principle in probability that describes how to update the probability of a hypothesis based on new evidence. It relates the conditional probabilities of two events and is expressed as P(A|B) = [P(B|A) * P(A)] / P(B). This theorem helps in understanding the relationship between P(M|B) and P(B|M).
Independence of Events
Two events are considered independent if the occurrence of one does not affect the probability of the other. In this case, if M and B are independent, then P(M|B) would equal P(M), and P(B|M) would equal P(B). Understanding whether M and B are independent is crucial for determining if P(M|B) is the same as P(B|M).
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