\(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\) both represent the occurrence of someevent, e.g. the number that comes up after rolling a fair die or if a person is a smoker or a non-smoker. \(\displaystyle{P}{\left({X}_{{1}}\right)}\) represents the probability that the event \(\displaystyle{X}_{{1}}\) occurs. \(\displaystyle{P}{\left(\frac{{Y}_{{2}}}{{X}_{{1}}}\right)}\) represents the probability that the event \(\displaystyle{Y}_{{2}}\) occurs given that the event \(\displaystyle{X}_{{1}}\) has occurred. In other words, we wantto know what chance \(\displaystyle{Y}_{{2}}\) will occur knowing that \(\displaystyle{X}_{{1}}\) has already happened.

The joint probability of \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\), denoted by \(\displaystyle{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}\) or \(\displaystyle{P}{\left({Y}_{{2}}\cap{X}_{{1}}\right)}\)

\(\displaystyle{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}={P}{\left({Y}_{{2}}\cap{X}_{{1}}\right)}\)

by the commutative property, representsthe probability that both events \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\) occur.

Use the following identity to compute

\(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}={\frac{{{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}}}{{{P}{\left({X}_{{1}}\right)}}}}\)

The formula for \(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}\) can be derived in the following manner: If the event \(\displaystyle{X}_{{1}}\) occurs, then in order for the event \(\displaystyle{Y}_{{2}}\) to occur, then they both have to occur. This explains the numerator. Since we know that the event \(\displaystyle{X}_{{1}}\) has occurred and we are only interested in a small part of \(\displaystyle{X}_{{1}}\) (case where \(\displaystyle{Y}_{{2}}\) occurs when \(\displaystyle{X}_{{1}}\) has occured), then \(\displaystyle{X}_{{1}}\) represents the set of all possibleoutcomes. This explains the denominator.

The joint probability of \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\), denoted by \(\displaystyle{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}\) or \(\displaystyle{P}{\left({Y}_{{2}}\cap{X}_{{1}}\right)}\)

\(\displaystyle{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}={P}{\left({Y}_{{2}}\cap{X}_{{1}}\right)}\)

by the commutative property, representsthe probability that both events \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\) occur.

Use the following identity to compute

\(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}={\frac{{{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}}}{{{P}{\left({X}_{{1}}\right)}}}}\)

The formula for \(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}\) can be derived in the following manner: If the event \(\displaystyle{X}_{{1}}\) occurs, then in order for the event \(\displaystyle{Y}_{{2}}\) to occur, then they both have to occur. This explains the numerator. Since we know that the event \(\displaystyle{X}_{{1}}\) has occurred and we are only interested in a small part of \(\displaystyle{X}_{{1}}\) (case where \(\displaystyle{Y}_{{2}}\) occurs when \(\displaystyle{X}_{{1}}\) has occured), then \(\displaystyle{X}_{{1}}\) represents the set of all possibleoutcomes. This explains the denominator.