Answer :
Certainly! Let's go through the problem step-by-step:
### Part (a): Using Conditional Probability
1. Total Chips and Defective Chips:
- Total chips: 5000
- Defective chips: 20
2. Probability of the First Chip Being Defective:
The probability that the first chip selected is defective is:
[tex]\[ P(\text{First Defective}) = \frac{\text{Number of Defective Chips}}{\text{Total Number of Chips}} = \frac{20}{5000} \][/tex]
3. Updating After the First Selection:
If the first chip selected is defective, then:
- Remaining total chips: 4999
- Remaining defective chips: 19
4. Probability of the Second Chip Being Defective Given the First Was Defective:
The probability that the second chip selected is defective, given that the first chip was defective, is:
[tex]\[ P(\text{Second Defective} \mid \text{First Defective}) = \frac{\text{Remaining Defective Chips}}{\text{Remaining Total Chips}} = \frac{19}{4999} \][/tex]
5. Combined Probability Using Conditional Probability:
Using conditional probability, the combined probability that both chips are defective is given by:
[tex]\[ P(\text{Both Defective}) = P(\text{First Defective}) \times P(\text{Second Defective} \mid \text{First Defective}) = \left( \frac{20}{5000} \right) \times \left( \frac{19}{4999} \right) \][/tex]
This evaluates to:
[tex]\[ P(\text{Both Defective}) \approx 1.52 \times 10^{-5} \][/tex]
Hence, the probability that two randomly selected chips are defective using conditional probability is approximately:
[tex]\[ 0.00001520 \][/tex]
### Part (b): Assuming Independent Events
1. Probability of the First Chip Being Defective:
The probability that the first chip selected is defective is the same as before:
[tex]\[ P(\text{First Defective}) = \frac{20}{5000} = 0.004 \][/tex]
2. Assuming Independence for the Second Selection:
If events are independent, the probability that the second chip is defective is unaffected by the first selection. Thus:
[tex]\[ P(\text{Second Defective, Independent}) = \frac{20}{5000} = 0.004 \][/tex]
3. Combined Probability Under Independence Assumption:
The combined probability under the assumption of independent events is:
[tex]\[ P(\text{Both Defective, Independent}) = P(\text{First Defective}) \times P(\text{Second Defective, Independent}) = 0.004 \times 0.004 = 0.000016 \][/tex]
Hence, the probability that two randomly selected chips are defective under the assumption of independent events is:
[tex]\[ 0.000016 \][/tex]
### Final Answer
- (a) The probability that two randomly selected chips are defective using conditional probability is approximately [tex]\(0.00001520\)[/tex].
- (b) The probability that two randomly selected chips are defective under the assumption of independent events is [tex]\(0.000016\)[/tex].
### Part (a): Using Conditional Probability
1. Total Chips and Defective Chips:
- Total chips: 5000
- Defective chips: 20
2. Probability of the First Chip Being Defective:
The probability that the first chip selected is defective is:
[tex]\[ P(\text{First Defective}) = \frac{\text{Number of Defective Chips}}{\text{Total Number of Chips}} = \frac{20}{5000} \][/tex]
3. Updating After the First Selection:
If the first chip selected is defective, then:
- Remaining total chips: 4999
- Remaining defective chips: 19
4. Probability of the Second Chip Being Defective Given the First Was Defective:
The probability that the second chip selected is defective, given that the first chip was defective, is:
[tex]\[ P(\text{Second Defective} \mid \text{First Defective}) = \frac{\text{Remaining Defective Chips}}{\text{Remaining Total Chips}} = \frac{19}{4999} \][/tex]
5. Combined Probability Using Conditional Probability:
Using conditional probability, the combined probability that both chips are defective is given by:
[tex]\[ P(\text{Both Defective}) = P(\text{First Defective}) \times P(\text{Second Defective} \mid \text{First Defective}) = \left( \frac{20}{5000} \right) \times \left( \frac{19}{4999} \right) \][/tex]
This evaluates to:
[tex]\[ P(\text{Both Defective}) \approx 1.52 \times 10^{-5} \][/tex]
Hence, the probability that two randomly selected chips are defective using conditional probability is approximately:
[tex]\[ 0.00001520 \][/tex]
### Part (b): Assuming Independent Events
1. Probability of the First Chip Being Defective:
The probability that the first chip selected is defective is the same as before:
[tex]\[ P(\text{First Defective}) = \frac{20}{5000} = 0.004 \][/tex]
2. Assuming Independence for the Second Selection:
If events are independent, the probability that the second chip is defective is unaffected by the first selection. Thus:
[tex]\[ P(\text{Second Defective, Independent}) = \frac{20}{5000} = 0.004 \][/tex]
3. Combined Probability Under Independence Assumption:
The combined probability under the assumption of independent events is:
[tex]\[ P(\text{Both Defective, Independent}) = P(\text{First Defective}) \times P(\text{Second Defective, Independent}) = 0.004 \times 0.004 = 0.000016 \][/tex]
Hence, the probability that two randomly selected chips are defective under the assumption of independent events is:
[tex]\[ 0.000016 \][/tex]
### Final Answer
- (a) The probability that two randomly selected chips are defective using conditional probability is approximately [tex]\(0.00001520\)[/tex].
- (b) The probability that two randomly selected chips are defective under the assumption of independent events is [tex]\(0.000016\)[/tex].
