Conditional probabilities for A, given B

Sometimes we are interested in situations where we have partial knowledge about the outcome of an experiment. Information from this partial knowledge is expressed via conditional probability.

Definition

The conditional probability that an event \(A\) occurs, given that \(B\) is known to have occurred is

\[ P(A \mid B) = \frac {P(A \textbf { and } B)} {P(B)} \]

Consider the hair colour and eye colour of one person selected at random from a group of \(n\) teenagers. If \(n_{\text{blue}}\) of the \(n\) teenagers have blue eyes and \(n_{\text{blue,blonde}}\) both have blue eyes and are blonde, the conditional probability of the teenager being blonde, given having blue eyes is

\[ P(\text {blonde} \mid \text {blue eyes}) = \frac {P(\text {blue eyes} \textbf { and } \text {blonde})} {P(\text {blonde})} = \frac { \frac {n_{\text {blue, blonde}}} {n}} { \frac {n_{\text {blue}}} {n}} = \frac { n_{\text {blue, blonde}}} {n_{\text {blue}}} \]

In other words, the definition gives the proportion of blue-eyed teenagers who are also blonde.

Conditional probabilities as a rescaling of joint probabilities

If the probabilities for possible values of hair colour, \(Y\), and eye colour, \(X\), are laid out in a table — their joint probabilities — then the conditional probabilities for \(Y\), given \(X = x\), can be found by rescaling that row of the table (dividing the whole row by \(p_x\)). This makes the row sum to 1.0, as shown in the diagram below.

Two sets of conditional probabilities

Note that there is an equivalent formula for conditional probability of \(B\) given that \(A\) has occurred.

\[ P(B \mid A) = \frac {P(A \textbf { and } B)} {P(A)} \]

You should be careful to distinguish between \(P(A \mid B)\) and \(P(B \mid A)\). They have different interpretations and usually also have different values.

The probability of being pregnant, given that a randomly selected person is female would be fairly small. The probability of being female, given that a person is pregnant is 1.0 !!