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.
is p(m|b) the same as p(b|m)?
a. no, 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 someone with blue eyes, given that a male 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. yes, because p(b|m) represents the probability of getting a male, given that someone with blue eyes has been selected

Explanation:

First Question (What does \( P(M|B) \) represent?):

In conditional probability, \( P(X|Y) \) 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. Option A is \( P(B|M) \) (swapped events), Option C is \( P(M \cup B) \), Option D is \( P(M \cap B) \). So the correct option is B.

\( P(M|B) \) is \( P(\text{Male}|\text{Blue eyes}) \), and \( P(B|M) \) is \( P(\text{Blue eyes}|\text{Male}) \). These are different conditional probabilities (different "given" conditions and different events being measured). Option A correctly states that \( P(B|M) \) is the probability of blue eyes given a male is selected, so they are not the same. Option B says yes (incorrect), Option C misdefines \( P(B|M) \), Option D is incorrect.

Answer:

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

Second Question (Is \( P(M|B) \) the same as \( P(B|M) \)?):