Question

describing conditional probabilities
explain why ( p(a|d) ) and ( p(d|a) ) from the table below are not equal.

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\). For \(P(A|D)\), \(P(A|D)=\frac{n(A\cap D)}{n(D)}\) (where \(n\) represents the number of elements). From the table, \(n(A\cap D) = 2\) and \(n(D)=10\). So \(P(A|D)=\frac{2}{10}=\frac{1}{5}\).

Step2: Calculate \(P(D|A)\)

Using the formula \(P(D|A)=\frac{n(A\cap D)}{n(A)}\). From the table, \(n(A\cap D) = 2\) and \(n(A)=8\). So \(P(D|A)=\frac{2}{8}=\frac{1}{4}\).

Answer:

Since \(P(A|D)=\frac{1}{5}\) and \(P(D|A)=\frac{1}{4}\), and \(\frac{1}{5}
eq\frac{1}{4}\), \(P(A|D)\) and \(P(D|A)\) are not equal. The reason is that in the formula for conditional probability \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\), the denominators \(P(Y)\) (in \(P(A|D)\) the denominator is \(P(D)\) and in \(P(D|A)\) the denominator is \(P(A)\)) are different.