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

Explanation:

In probability, the notation \(P(A|B)\) represents the conditional probability of event \(A\) occurring given that event \(B\) has already occurred. Here, \(M\) is the event of selecting a male and \(B\) is the event of selecting someone with blue eyes. So, \(P(M|B)\) means the probability of getting a male given that someone with blue eyes has been selected.

For the second part, \(P(M|B)=\frac{P(M\cap B)}{P(B)}\) and \(P(B|M)=\frac{P(M\cap B)}{P(M)}\). Since \(P(B)\) (probability of blue - eyed people) and \(P(M)\) (probability of male people) are generally not equal (unless the proportion of blue - eyed people among all adults is the same as the proportion of male people among all adults), \(P(M|B)\) is not the same as \(P(B|M)\) in most cases.

Answer:

For the first question: D. The probability of getting a male, given that someone with blue eyes has been selected.

For the second question: No, \(P(M|B)\) is not the same as \(P(B|M)\) because \(P(M|B)=\frac{P(M\cap B)}{P(B)}\) and \(P(B|M)=\frac{P(M\cap B)}{P(M)}\), and \(P(B)
eq P(M)\) in general.