Question

let (a) and (b) be two events such that (p(a) = 0.80) and (p(b) = 0.08).

do not round your responses. (if necessary, consult a list of formulas.)

(a) determine (p(a \cup b)), given that (a) and (b) are independent.

(b) determine (p(a \cup b)), given that (a) and (b) are mutually exclusive.

Explanation:

Identify the given probabilities

We are given the probabilities of two events \(A\) and \(B\):

$$ P(A) = 0.80,\quad P(B) = 0.08 $$

Calculate the union for independent events

Using the Independent Events Probability knowledge point, if \(A\) and \(B\) are independent, their intersection is:

$$ P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.08 = 0.064 $$

Using the Addition Rule of Probability knowledge point, we find the union:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.80 + 0.08 - 0.064 = 0.816 $$

Calculate the union for mutually exclusive events

If \(A\) and \(B\) are mutually exclusive, they cannot occur at the same time, meaning:

$$ P(A \cap B) = 0 $$

Using the Addition Rule of Probability knowledge point, the union simplifies to:

$$ P(A \cup B) = P(A) + P(B) = 0.80 + 0.08 = 0.88 $$

Answer:

Question a

(a) Determine \(P(A \cup B)\), given that \(A\) and \(B\) are independent.
<blank>0.816</blank>

Question b

(b) Determine \(P(A \cup B)\), given that \(A\) and \(B\) are mutually exclusive.
<blank>0.88</blank>