Math Snap

STEP 1

1. The tree diagram represents a probability experiment with two trials.
2. Probabilities for the first trial are given as $P(A) = 0.4$, $P(B) = 0.3$, and $P(C) = 0.2$.
3. Conditional probabilities for the second trial are given for each branch.

STEP 2

1. Verify and calculate $P(A)$.
2. Calculate $P(E \mid A)$.
3. Calculate $P(A \cap E)$.
4. Calculate $P(E)$.

STEP 3

Verify the probability of event $A$.
Given: $P(A) = 0.4$
Since the probability is directly provided in the tree diagram, no further calculation is needed.

STEP 4

Calculate the conditional probability $P(E \mid A)$.
From the tree diagram, the probability of $E$ given $A$ is:
$$P(E \mid A) = 0.5$$

STEP 5

Calculate the joint probability $P(A \cap E)$.
Using the formula for joint probability:
$$P(A \cap E) = P(A) \times P(E \mid A)$$ Substitute the known values:
$$P(A \cap E) = 0.4 \times 0.5 = 0.2$$

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) $P(E) = 0.37$