Solution to Question 4:

The problem asks for the probability that a randomly selected label contains a defective compact disk.

1. Determine the contribution from each manufacturer:

   • Manufacturer I: $0.37$ (supply probability) × $0.07$ (defective rate) = $0.0259$
   • Manufacturer II: $0.45$ × $0.05$ = $0.0225$
   • Manufacturer III: $0.18$ × $0.04$ = $0.0072$

2. Add these contributions together: $P(\text{Defective}) = 0.0259 + 0.0225 + 0.0072 = 0.0556$

Thus, the probability that a randomly selected label contains a defective compact disk is $0.0556$.
The correct answer is: 0.0556.

Solution to Question 5:

We need to calculate the probability that the pink jar was chosen given that the cookie selected is a sugar cookie.

1. Step 1: Calculate total probabilities for sugar cookies from each jar:

   • Blue jar: $P(\text{Blue}) = \frac{1}{2}$, $P(\text{Sugar | Blue}) = \frac{12}{20}$, so $P(\text{Sugar and Blue}) = \frac{1}{2} \times \frac{12}{20} = \frac{6}{20} = 0.3$.
   • Green jar: $P(\text{Green}) = \frac{1}{4}$, $P(\text{Sugar | Green}) = \frac{10}{22}$, so $P(\text{Sugar and Green}) = \frac{1}{4} \times \frac{10}{22} = \frac{10}{88} = 0.1136$.
   • Pink jar: $P(\text{Pink}) = \frac{1}{4}$, $P(\text{Sugar | Pink}) = \frac{13}{42}$, so $P(\text{Sugar and Pink}) = \frac{1}{4} \times \frac{13}{42} = \frac{13}{168} = 0.0774$.

2. Step 2: Total probability of selecting a sugar cookie: $P(\text{Sugar}) = 0.3 + 0.1136 + 0.0774 = 0.491$

3. Step 3: Use Bayes’ theorem to find $P(\text{Pink | Sugar})$: $P(\text{Pink | Sugar}) = \frac{P(\text{Sugar and Pink})}{P(\text{Sugar})} = \frac{0.0774}{0.491} \approx 0.1576$

Thus, the probability that the pink jar was chosen given that the cookie is a sugar cookie is $0.1576$.
The correct answer is: 0.1576.

Follow-Up Questions:

1. Can you explain the difference between conditional and unconditional probabilities?
2. How would the probabilities change in Question 5 if the pink jar had an additional sugar cookie?
3. What is Bayes' theorem and how is it applied to real-world situations?
4. If Manufacturer III’s defect rate changes, how would that affect the result in Question 4?
5. Can you calculate the expected number of defective CDs if 1,000 CDs are ordered?

Tip: For problems involving probability trees, visualizing the branches with their probabilities helps clarify the relationships.