Question

the diagram below represents an experiment consisting of two independent trials. find p(a and d).

\\(p(a \text{ and } d) = \frac{?}{\quad}\\)

simplify your answer completely.

Explanation:

Identify path probabilities

The probability of selecting branch \(A\) is:

$$P(A) = \frac{1}{3}$$

The conditional probability of selecting branch \(D\) given \(A\) is:

$$P(D|A) = \frac{1}{4}$$

Calculate joint probability

Using the multiplication rule for dependent/sequential events along a branch:

$$P(A \text{ and } D) = P(A) \times P(D|A)$$

$$P(A \text{ and } D) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$$

Answer:

The diagram below represents an experiment consisting of two independent trials. Find P(A and D).

\(P(A \text{ and } D) =\) <blank>\(\frac{1}{12}\)</blank>

Simplify your answer completely.