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The possibility of an event or outcome occurring based on the occurrence of a preceding event or outcome is known as conditional probability. The updated probability of the subsequent, or conditional, event is multiplied by the probability of the preceding, or conditional, event to get conditional probability.

We're given events A and B: "the person is above \(6\) feet tall" and "the person is a professional basketball player."

We don't have any additional information, so we'll have to "guess."

The conditional probability of A given that the event \(B\) has occurred, for which \(P(B) > 0\), is

\(P(A\mid B) = \frac{{P(A \cap B)}}{{P(B)}}\)

for any two events \(A\) and \(B\).

There are professional basketball players as well as persons taller than six feet, therefore the likelihood for both events is clearly greater than zero.

The probability of those events is calculated as

\(P(A) = \frac{{\# {\rm{ of favorable outcomes in }}A}}{{\# {\rm{ of outcomes in the sample space }}}}\)

However, because we don't know how many people are taller than six feet or how many professional basketball players there are in the United States, we can either look for and use statistical data or simply conclude that there are more people taller than six feet than professional basketball players (most of the basketball players are taller than 6 feet, however, not most of the \(6\) feet people are professional basketball players).

Therefore

\(N(A) > N(B)\)

The number of favorable outcomes in event A is bigger than the number of favorable outcomes in B which indicates that

\(\begin{array}{c}P(A) =& \frac{{\# {\rm{ of favorable outcomes in A }}}}{{\# {\rm{ of outcomes in the sample space }}}}\\ =& \frac{{N(A)}}{N} > \frac{{N(B)}}{N}\\ =& P(B).\end{array}\)

Using this, we have that

\(P(A) > P(B)\;\;\; \Rightarrow \;\;\;\frac{1}{{P(A)}} < \frac{1}{{P(B)}}\)

and now, because \(P(A \cap B) > 0\)(there are six+ feet professional basketball players), we have

\(\frac{{P(A \cap B)}}{{P(A)}} < \frac{{P(A \cap B)}}{{P(B)}}\)

or equally,

\(P(B\mid A) < P(A\mid B)\).