Let's begin by using the given information to calculate the answers for questions e - g.

Given Information:

• $P(A) = 0.47$
• $P(B) = 0.5$
• $A$ and $B$ are dependent.
• $P(A \mid B) = 0.468$

Since $A$ and $B$ are dependent, we can use the conditional probability formula to find the joint probability.

e) Find $P(A \cap B)$

The conditional probability formula is:

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

We need to solve for $P(A \cap B)$:

$P(A \cap B) = P(A \mid B) \cdot P(B)$

Substituting the given values:

$P(A \cap B) = 0.468 \times 0.5 = 0.234$

Answer: $P(A \cap B) = 0.23400$ (rounded to 5 decimal places)

f) Find $P(A \cup B)$

The formula for the union of two events is:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Substituting the known values:

$P(A \cup B) = 0.47 + 0.5 - 0.234$

$P(A \cup B) = 0.736$

Answer: $P(A \cup B) = 0.73600$ (rounded to 5 decimal places)

g) Find $P(B \mid A)$

The conditional probability of $B$ given $A$ is given by:

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

Substituting the known values:

$P(B \mid A) = \frac{0.234}{0.47}$

$P(B \mid A) = 0.49787$

Answer: $P(B \mid A) = 0.49787$ (rounded to 5 decimal places)

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Do you have any questions or need further details?

Related Questions:

1. How would you calculate $P(A \cap B)$ if $A$ and $B$ were independent?
2. What is the difference between dependent and independent events?
3. How can the union formula be used for mutually exclusive events?
4. How would you determine $P(A^c \cap B)$, the probability that $A$ does not occur but $B$ does?
5. How do conditional probabilities change if $P(B) = 0$?

Tip:

Always remember that when dealing with dependent events, conditional probabilities provide a way to understand how the occurrence of one event affects the likelihood of another.