Y and Z are independent

Question

Y and Z are independent events.

Part (a)

Rewrite the basic Addition Rule P(Y OR Z) = P(Y) + P(Z) − P(Y AND Z) using the information that Y and Z are independent events.

A. P(Y OR Z) = P(Y) + P(Z)

B. P(Y OR Z) = P(Y) + P(Z) − P(Z)P(Z|Y)

C. P(Y OR Z) = P(Y) + P(Z) − P(Y)P(Z)

D. P(Y OR Z) = P(Y) + P(Z) + P(Y)P(Z)

E. P(Y OR Z) = P(Y) + P(Z) − P(Y)P(Y|Z)

Part (b)

Use the rewritten rule to find P(Z) if P(Y OR Z) = 0.67 and P(Y) = 0.44. (Round your answer to two decimal places.)

P(Z) =_____

Answer 1

Part A

Given:

We use the definition of the joint probability of and , : 

Upon substitution:

Answer is C

Part B

Given:

Gathering all the terms that have in them on one side of the equation (i.e., subtracting from both sides):

Factoring out the from both terms on the right-hand side:

Isolating by dividing both sides by :

Plugging in the numerical values given above:

Answer 2

P(Y or Z) = P(Y) + P(Z) - P(Y and Z)

= P(Y) + P(Z) - [P(Y)*P(Z)]

Y and Z are independent events if P(Y and Z) = P(Y)*P(Z). Therefore, the answer is Choice C.

Let P(Y or Z) = 0.67 and P(Y) = 0.44.

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

0.23 = P(Z) - 0.44*P(Z)

0.23 = 0.56*P(Z)

P(Z) = 0.41