1. **Problem Statement:** Destiny samples 4 products without replacement from a batch of 100 products, 14 of which are defective. We need to find the probabilities for three cases: 2. **Formulas and Important Rules:** - The total number of ways to choose 4 products from 100 is given by the combination formula $$\binom{100}{4}$$. - The number of ways to choose defective or non-defective products is also calculated using combinations. - Probability is calculated as $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$. 3. **Part (a): Probability all 4 products are defective** - Number of ways to choose 4 defective products from 14 defective products: $$\binom{14}{4}$$ - Total ways to choose any 4 products from 100: $$\binom{100}{4}$$ - Probability: $$ P(\text{all defective}) = \frac{\binom{14}{4}}{\binom{100}{4}} $$ - Calculate combinations: $$ \binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001 $$ $$ \binom{100}{4} = \frac{100 \times 99 \times 98 \times 97}{4 \times 3 \times 2 \times 1} = 3921225 $$ - So, $$ P = \frac{1001}{3921225} \approx 0.000255 $$ - Rounded to 2 decimal places: **0.00** 4. **Part (b): Probability none of the products are defective** - Number of non-defective products: $$100 - 14 = 86$$ - Number of ways to choose 4 non-defective products from 86: $$\binom{86}{4}$$ - Probability: $$ P(\text{none defective}) = \frac{\binom{86}{4}}{\binom{100}{4}} $$ - Calculate $$\binom{86}{4}$$: $$ \binom{86}{4} = \frac{86 \times 85 \times 84 \times 83}{4 \times 3 \times 2 \times 1} = 2042970 $$ - So, $$ P = \frac{2042970}{3921225} \approx 0.52 $$ - Rounded to 2 decimal places: **0.52** 5. **Part (c): Probability at least one product is defective** - This is the complement of none defective: $$ P(\text{at least one defective}) = 1 - P(\text{none defective}) $$ - Substitute the value: $$ P = 1 - 0.52 = 0.48 $$ - Rounded to 2 decimal places: **0.48** **Final answers:** - a) 0.00 - b) 0.52 - c) 0.48