Question

Suppose a manufacturer finds that 95% of their production is normal but the final 5% has one or more flaws. Each flawed good has a financial cost to the firm of $1. The manufacturer tasks a team led by you, the manager to identify the likelihood of upcoming flaws by providing your best expectation. Here is what you found by reviewing past production.Of the flawed goods:80% show Type 1 flaws.50% show Type 2 flaws.30% show both Type 1 and Type 2 flaws.1. Make a Bivariate Table using the above data.2. What is the probability and expectations of type 1 flaw?3. What is the probability and expectation of Type 2 flaw?4. What is the probability and expectations of Type 1 and 2 flaws?5. What is the probability and expectations of no flaws?

Answer 1

1)

FLAW TYPE2 NO TYPE2 FLAW

TYPE1 0.015 0.025

NO TYPE1 FLAW 0.01 0.95

2) 0.04 and $0.04

3) 0.025 and $0.025

4) 0.015 and $0.015

5) 0.95 and $0.95

Explanation:

Given that;

financial cost = $1

p(flaw) = 0.05

p(type 1 flaw / flaw) = 80% = 0.8

p(type 2 flaw / flaw) = 50% = 0.5

p( type 1 and 2 flaw/flaw) = 30% = 0.30

1) Bivariate Table

p( type 1 flaw) = p(flaw) × p(type 1 flaw/flaw) = 0.05 × 0.8 = 0.04

p( type 2 flaw) = p(flaw) × p(type 2 flaw/flaw) = 0.05 × 0.5 = 0.025

p( type 1 and 2 flaw) = p(flow) × p( type 1 & 2 flaw/flaw) = 0.05 × 0.3 = 0.015

p( only 1 flow) = 0.04 - 0.015 = 0.025

p( only 2 flow) = 0.025 - 0.015 = 0.01

THEREFORE the Bivariate Table;

FLAW TYPE2 NO TYPE2 FLAW

TYPE1 0.015 0.025

NO TYPE1 FLAW 0.01 0.95

2) probability and expectations of type 1 flaw?

p( type 1 flaw) = p(flaw) × p(type 1 flaw/flaw) = 0.05 × 0.8 = 0.04

Expected financial cost to the firm per good = $1 × 0.04 = $0.04

3) probability and expectation of Type 2 flaw

p( type 2 flaw) = p(flaw) × p(type 2 flaw/flaw) = 0.05 × 0.5 = 0.025

Expected financial cost to the firm per good = $1 × 0.025 = $0.025

4) probability and expectations of Type 1 and 2 flaws

p( type 1 and 2 flaw) = p(flow) × p( type 1 & 2 flaw/flaw) = 0.05 × 0.3 = 0.015

Expected financial cost to the firm per good = $1 * 0.015 = $0.015

5) probability and expectations of no flaws?

Probability of no flaw = P(No flaw) =95% = 0.95

Expected financial cost saved the firm per good due to no flaw

= $1 × 0.95 = $0.95