Questions: 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)? What does P(M B) represent? A. The probability of getting a male, given that someone with blue eyes has been selected B. The probability of getting a male and getting someone with blue eyes C. The probability of getting someone with blue eyes, given that a male has been selected D. The probability of getting a male or getting someone with blue eyes Is P(M B) the same as P(B M)? A. Yes, because P(B M) represents the probability of getting someone with blue eyes, given that a male has been selected B. Yes, because P(B M) represents the probability of getting a male, given that someone with blue eyes has been selected C. No, because P(B M) represents the probability of getting a male, given that someone with blue eyes has been selected D. No, because P(B M) represents the probability of getting someone with blue eyes, given that a male has been selected

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)? What does P(M B) represent? A. The probability of getting a male, given that someone with blue eyes has been selected B. The probability of getting a male and getting someone with blue eyes C. The probability of getting someone with blue eyes, given that a male has been selected D. The probability of getting a male or getting someone with blue eyes Is P(M B) the same as P(B M)? A. Yes, because P(B M) represents the probability of getting someone with blue eyes, given that a male has been selected B. Yes, because P(B M) represents the probability of getting a male, given that someone with blue eyes has been selected C. No, because P(B M) represents the probability of getting a male, given that someone with blue eyes has been selected D. No, because P(B M) represents the probability of getting someone with blue eyes, given that a male has been selected
Transcript text: 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 $P(M \mid B)$ represent? is $P(M \mid B)$ the same as $\mathrm{P}(\mathrm{B} \mid \mathrm{M})$ ? What does $\mathrm{P}(\mathrm{M} \mid \mathrm{B})$ represent? A. The probability of getting a male, given that someone with blue eyes has been selected B. The probability of getting a male and getting someone with blue eyes C. The probability of getting someone with blue eyes, given that a male has been selected D. The probability of getting a male or getting someone with blue eyes Is $P(M \mid B)$ the same as $P(B \mid M)$ ? A. Yes, because $P(B \mid M)$ represents the probability of getting someone with blue eyes, given that a male has been selected B. Yes, because $P(B \mid M)$ represents the probability of getting a male, given that someone with blue eyes has been selected C. No, because $P(B \mid M)$ represents the probability of getting a male, given that someone with blue eyes has been selected D. No, because $P(B \mid M)$ represents the probability of getting someone with blue eyes, given that a male has been selected
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Solution

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Solution Steps

Solution Approach
  1. The expression \( P(M \mid B) \) represents the conditional probability of selecting a male given that the person selected has blue eyes. Therefore, the correct interpretation is option A.
  2. To determine if \( P(M \mid B) \) is the same as \( P(B \mid M) \), we need to understand that these are two different conditional probabilities. \( P(M \mid B) \) is the probability of selecting a male given blue eyes, while \( P(B \mid M) \) is the probability of selecting someone with blue eyes given that the person is male. Therefore, the correct answer is option D.
Step 1: Understanding \( P(M \mid B) \)

The expression \( P(M \mid B) \) represents the conditional probability of selecting a male (\( M \)) given that the selected individual has blue eyes (\( B \)). This can be interpreted as the likelihood of choosing a male from the subset of individuals who have blue eyes. Therefore, the correct interpretation is:

\[ P(M \mid B) = \text{The probability of getting a male, given that someone with blue eyes has been selected} \]

Thus, the answer is A.

Step 2: Comparing \( P(M \mid B) \) and \( P(B \mid M) \)

To determine if \( P(M \mid B) \) is the same as \( P(B \mid M) \), we analyze the definitions of both probabilities.

  • \( P(M \mid B) \) is the probability of selecting a male given that the individual has blue eyes.
  • \( P(B \mid M) \) is the probability of selecting someone with blue eyes given that the individual is male.

Since these two probabilities represent different conditions, they are not equal. Therefore, the correct answer is:

\[ P(M \mid B) \neq P(B \mid M) \]

Thus, the answer is D.

Final Answer

The answers to the questions are:

  1. The answer to the first question is A.
  2. The answer to the second question is D.

\[ \boxed{\text{1. A, 2. D}} \]

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