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Let's break down the problem step by step.

### Given Information:
1. **Modes of Transport**:
   - 50% by land
     - Of this 50%, 40% is via highway, and the remaining 60% is by rail.
   - 30% by sea.
   - 20% by air.

2. **Damaged Cargo Percentages**:
   - 10% by highway.
   - 5% by rail.
   - 6% by sea.
   - 2% by air.

### Part (a): What percentage of all cargoes may be expected to be damaged?

We will calculate the expected damage by multiplying the percentage of goods transported by each mode and the corresponding damage rates. 

- **Land Transport (50%)**:
  - 40% of land is via highway → \( 50\% \times 40\% = 20\% \) transported by highway.
  - 60% of land is via rail → \( 50\% \times 60\% = 30\% \) transported by rail.
  
  Now, calculate the damaged cargo for land transport:
  - Highway: \( 20\% \times 10\% = 2\% \)
  - Rail: \( 30\% \times 5\% = 1.5\% \)
  
  Total damaged from land: \( 2\% + 1.5\% = 3.5\% \).

- **Sea Transport (30%)**:
  - \( 30\% \times 6\% = 1.8\% \) damaged.

- **Air Transport (20%)**:
  - \( 20\% \times 2\% = 0.4\% \) damaged.

### Total Damaged Cargo:
Now, summing all the damages from land, sea, and air:

\[
\text{Total damaged} = 3.5\% + 1.8\% + 0.4\% = 5.7\%
\]

Thus, 5.7% of all cargo is expected to be damaged.

---

### Part (b): If a damaged cargo is received, what is the probability that it was shipped by land, by sea, or by air?

This part involves conditional probability. To find the probability that a damaged cargo came from a specific mode of transportation, we use the formula:

\[
P(\text{Transport Mode} | \text{Damaged}) = \frac{P(\text{Damaged} \cap \text{Transport Mode})}{P(\text{Damaged})}
\]

We already calculated the total damaged probability as \( 5.7\% \). Now, we'll calculate the conditional probabilities for land, sea, and air.

- **Land (Highway + Rail)**:
  \[
  P(\text{Land} | \text{Damaged}) = \frac{3.5\%}{5.7\%} \approx 0.614 \text{ or } 61.4\%
  \]
  
- **Sea**:
  \[
  P(\text{Sea} | \text{Damaged}) = \frac{1.8\%}{5.7\%} \approx 0.316 \text{ or } 31.6\%
  \]

- **Air**:
  \[
  P(\text{Air} | \text{Damaged}) = \frac{0.4\%}{5.7\%} \approx 0.07 \text{ or } 7\%
  \]

### Final Answers:
1. **Part (a)**: 5.7% of all cargoes are expected to be damaged.
2. **Part (b)**: 
   - Probability that the damaged cargo was shipped by land: **61.4%**.
   - Probability that the damaged cargo was shipped by sea: **31.6%**.
   - Probability that the damaged cargo was shipped by air: **7%**.

Let me know if you need more details or have any questions!

---

Here are 5 related questions to expand the understanding:
1. What is the total amount of goods transported by rail if 1000 shipments were sent?
2. If sea transport damage rate increased to 10%, how would this affect the overall damaged percentage?
3. What is the expected percentage of undamaged cargo for each mode of transport?
4. How would the probabilities change if air transport damage rates were reduced to 1%?
5. What is the conditional probability of receiving a damaged cargo if only rail transport is used?

### Tip:
Understanding conditional probabilities helps in decision-making, especially in logistics and risk assessment. Always compare the likelihood of an event happening given a condition versus the overall likelihood.

There are three modes of transporting material from New York to Florida, namely, by land, sea, or air. Also, land transportation may be by rail or highway. About half of the materials are transported by land, 30% by sea, and the rest by air. Also, 40% of all land transportation is by highway and the rest by rail shipments. The percentages of damaged cargo are, respectively, 10% by highway, 5% by rail, 6% by sea, and 2% by air. (a) What percentage of all cargoes may be expected to be damaged? (b) If a damaged cargo is received, what is the probability that it was shipped by land? By sea? By air?

Explore how to calculate the expected percentage of damaged cargo when transporting materials by land, sea, and air. Learn to compute conditional probabilities for damaged cargo based on the mode of transport using real-world scenarios.

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