QuestionThe accompanying tree diagram represents an experiment consisting of two trials.
Use the diagram to find the probabilities below. $$\begin{array}{l} \text { (a) } P(A) \\ 1.4 \end{array}$$ $$\begin{array}{l} \text { (b) } \quad P(E \mid A) \\ 5 \\ \text { (c) } \quad P(A \cap E) \\ 15 \end{array}$$ (d) $P(E)$ $$35$$

Studdy Solution
Calculate the probability of $E$, $P(E)$.
Using the law of total probability:
$$P(E) = P(A \cap E) + P(B \cap E) + P(C \cap E)$$
Calculate each term:
$$P(B \cap E) = P(B) \times P(E \mid B) = 0.3 \times 0.3 = 0.09$$
$$P(C \cap E) = P(C) \times P(E \mid C) = 0.2 \times 0.4 = 0.08$$
Add the probabilities:
$$P(E) = 0.2 + 0.09 + 0.08 = 0.37$$
The probabilities are: (a) $P(A) = 0.4$ (b) $P(E \mid A) = 0.5$ (c) $P(A \cap E) = 0.2$ (d) $P(E) = 0.37$