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 $P\left(M|B\right)$represent? Is$P\left(M|B\right)$ the same as$P\left(B|M\right)$ ?

Short answer

$P\left(M|B\right)$represents the probability of selecting a male given that he has blue eyes.

$P\left(B|M\right)$represents the probability of selecting a person with blue eyes, given that he is a male.

No,$P\left(M|B\right)$ is not the same as $P\left(B|M\right)$.

Step-by-step solution

Given information

Two events are defined as follows.

• M is the event of selecting a person who is male.
• B is the event of selecting a person who has blue eyes.

Concept of conditional probability

For any two events A and B, the conditional probability of that event A occurs, given event B has already taken place, is denoted by $P\left(A|B\right)$. It is the probability of occurrence of event A, given that event B has already occurred.

Similarly, the conditional probability of that event B occurs, given event A has already taken place, is denoted by $P\left(B|A\right)$. It is the probability of occurrence of event A, given that event B has already occurred.

These are calledconditional probabilities, as each probability is associated with a condition.

Interpret the two conditional probabilities in terms of given events

Here, M is the event of selecting a male person, and B is the event of selecting a person with blue eyes.

$P\left(M|B\right)$is the probability of selecting a male person given that he has blue eyes.

$P\left(B|M\right)$is the probability of selecting a person with blue eyes given that he is a male.

Both probabilities stand for two different events (base condition as well as the event). Therefore,$P\left(M|B\right)$ is not the same as $P\left(B|M\right)$.