Question

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 someone with blue eyes, given that a male has been selected
b. the probability of getting a male, given that someone with blue eyes has been selected
c. the probability of getting a male or getting someone with blue eyes
d. the probability of getting a male and getting someone with blue eyes

Explanation:

To determine what \( P(M|B) \) represents, we use the definition of conditional probability. The notation \( P(X|Y) \) means the probability of event \( X \) occurring given that event \( Y \) has already occurred. Here, \( X = M \) (selecting a male) and \( Y = B \) (selecting someone with blue eyes). So \( P(M|B) \) is the probability of getting a male, given that someone with blue eyes has been selected.

Now let's analyze the options:

• Option A: This would be \( P(B|M) \), not \( P(M|B) \), so A is incorrect.
• Option B: Matches the definition of \( P(M|B) \), so B is correct.
• Option C: The probability of a male or blue eyes is \( P(M \cup B) \), not a conditional probability, so C is incorrect.
• Option D: The probability of a male and blue eyes is \( P(M \cap B) \), not a conditional probability, so D is incorrect.

For the second part (not fully shown in the question but about whether \( P(M|B) \) is the same as \( P(B|M) \)): In general, \( P(M|B)=\frac{P(M\cap B)}{P(B)} \) and \( P(B|M)=\frac{P(M\cap B)}{P(M)} \). These are equal only if \( P(B) = P(M) \), which is not generally true. So they are not the same. But for the first part (the multiple - choice question), we focus on the meaning of \( P(M|B) \).

Answer:

B. The probability of getting a male, given that someone with blue eyes has been selected