Question

Part (a) Rewrite the basic Addition Rule \(P(Y \text{ OR } Z) = P(Y) + P(Z) - P(Y \text{ AND } Z)\) using the information that Y and Z are independent events.

A. \(P(Y \text{ OR } Z) = P(Y) + P(Z)\)

B. \(P(Y \text{ OR } Z) = P(Y) + P(Z) - P(Z)P(Z|Y)\)

C. \(P(Y \text{ OR } Z) = P(Y) + P(Z) - P(Y)P(Z)\)

D. \(P(Y \text{ OR } Z) = P(Y) + P(Z) + P(Y)P(Z)\)

Part (b) Use the rewritten rule to find \(P(Z)\) if \(P(Y \text{ OR } Z) = 0.67\) and \(P(Y) = 0.44\).

A. \(0.89\)

B. \(0.23\)

C. \(0.33\)

D. \(0.56\)

Answer 1

The basic addition rule for probability when Y and Z are independent is P(Y OR Z) = P(Y) + P(Z) - P(Y)P(Z). After calculation, the answer for P(Z) is that it is 0.23.

Step-by-step explanation:

The student is asking about the addition rule for probability for independent events. When events Y and Z are independent, the probability that Y or Z will happen, denoted by P(Y OR Z), can be expressed as P(Y) + P(Z) - P(Y)P(Z). The independence of events allows us to multiply their individual probabilities to find the probability of both occurring together, which is the term P(Y AND Z) in the addition rule.

For part (a), because Y and Z are independent, the basic addition rule is modified to:

P(Y OR Z) = P(Y) + P(Z) - P(Y)P(Z)

For part (b), using the provided probabilities:

P(Y OR Z) = 0.67
P(Y) = 0.44

We plug those values into the equation:

0.67 = 0.44 + P(Z) - (0.44)(P(Z))

Solving for P(Z), we find that:

P(Z) = 0.23