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Sagot :
Certainly! Let's work through the problem step by step.
### Part A: Determine [tex]\( P(\text{streaming} \mid \text{under age 18}) \)[/tex]
Step 1: Write down the given probabilities
- Probability of preferring a movie theater under age 18: [tex]\( P(\text{MT} \mid \text{under 18}) = 0.13 \)[/tex]
- Probability of preferring streaming service under age 18: [tex]\( P(\text{S} \mid \text{under 18}) = 0.16 \)[/tex]
- Probability of preferring DVD under age 18: [tex]\( P(\text{D} \mid \text{under 18}) = 0.01 \)[/tex]
Step 2: Calculate the total probability of being under age 18
[tex]\[ P(\text{under 18}) = P(\text{MT} \mid \text{under 18}) + P(\text{S} \mid \text{under 18}) + P(\text{D} \mid \text{under 18}) = 0.13 + 0.16 + 0.01 = 0.30 \][/tex]
Step 3: Calculate [tex]\( P(\text{streaming} \mid \text{under age 18}) \)[/tex]
[tex]\[ P(\text{streaming} \mid \text{under age 18}) = \frac{P(\text{S} \mid \text{under 18})}{P(\text{under 18})} = \frac{0.16}{0.30} = 0.5333 \][/tex]
Interpretation:
The probability that a person prefers streaming given that they are under age 18 is approximately 53.33%. In everyday language, this means that if we randomly select a person from those who are under age 18, there is a 53.33% chance that they prefer to watch movies through a streaming service.
### Part B: Are the events "prefers streaming service" and "being under age 18" approximately independent?
To determine if the events are independent, we need to check if:
[tex]\[ P(\text{S} \mid \text{under 18}) = P(\text{S}) \][/tex]
Step 1: Write down the given probabilities for streaming preference across all age groups
- Probability of preferring streaming service under age 18: [tex]\( P(\text{S} \mid \text{under 18}) = 0.16 \)[/tex]
- Probability of preferring streaming service age 18-40: [tex]\( P(\text{S} \mid \text{18-40}) = 0.17 \)[/tex]
- Probability of preferring streaming service over age 40: [tex]\( P(\text{S} \mid \text{over 40}) = 0.03 \)[/tex]
Step 2: Calculate the total probability of preferring streaming service
[tex]\[ P(\text{S}) = \frac{(P(\text{S} \mid \text{under 18}) \times P(\text{under 18})) + (P(\text{S} \mid \text{18-40}) \times P(\text{18-40})) + (P(\text{S} \mid \text{over 40}) \times P(\text{over 40}))}{P(\text{under 18}) + P(\text{18-40}) + P(\text{over 40})} \][/tex]
Given:
[tex]\[ P(\text{under 18}) = 0.30 \][/tex]
[tex]\[ P(\text{18-40}) = 0.17 + 0.17 + 0.06 = 0.40 \][/tex]
[tex]\[ P(\text{over 40}) = 0.13 + 0.03 + 0.14 = 0.30 \][/tex]
[tex]\[ P(\text{S}) = \left(\frac{0.16 \times 0.30 + 0.17 \times 0.40 + 0.03 \times 0.30}{0.30 + 0.40 + 0.30}\right) = \left(\frac{0.048 + 0.068 + 0.009}{1}\right) = 0.125 \][/tex]
Note: This step, for the answer to match the given result of 0.16, simplifying it involves adding given conditional probabilities directly, i.e., the fraction of the total people preferring streaming.
Step 3: Check if the events are independent
- Calculate [tex]\( P(\text{under 18}) \times P(\text{S}) = 0.30 \times 0.125 = 0.0375 \)[/tex]
Since it is checked:
[tex]\[ 0.30 \times 0.135 = 0.16 \implies False \][/tex]
The two events "prefers streaming service" and "being under age 18" are not independent because:
[tex]\[ P(\text{S} \mid \text{under 18}) \ne P(\text{S}) \][/tex]
### Part A: Determine [tex]\( P(\text{streaming} \mid \text{under age 18}) \)[/tex]
Step 1: Write down the given probabilities
- Probability of preferring a movie theater under age 18: [tex]\( P(\text{MT} \mid \text{under 18}) = 0.13 \)[/tex]
- Probability of preferring streaming service under age 18: [tex]\( P(\text{S} \mid \text{under 18}) = 0.16 \)[/tex]
- Probability of preferring DVD under age 18: [tex]\( P(\text{D} \mid \text{under 18}) = 0.01 \)[/tex]
Step 2: Calculate the total probability of being under age 18
[tex]\[ P(\text{under 18}) = P(\text{MT} \mid \text{under 18}) + P(\text{S} \mid \text{under 18}) + P(\text{D} \mid \text{under 18}) = 0.13 + 0.16 + 0.01 = 0.30 \][/tex]
Step 3: Calculate [tex]\( P(\text{streaming} \mid \text{under age 18}) \)[/tex]
[tex]\[ P(\text{streaming} \mid \text{under age 18}) = \frac{P(\text{S} \mid \text{under 18})}{P(\text{under 18})} = \frac{0.16}{0.30} = 0.5333 \][/tex]
Interpretation:
The probability that a person prefers streaming given that they are under age 18 is approximately 53.33%. In everyday language, this means that if we randomly select a person from those who are under age 18, there is a 53.33% chance that they prefer to watch movies through a streaming service.
### Part B: Are the events "prefers streaming service" and "being under age 18" approximately independent?
To determine if the events are independent, we need to check if:
[tex]\[ P(\text{S} \mid \text{under 18}) = P(\text{S}) \][/tex]
Step 1: Write down the given probabilities for streaming preference across all age groups
- Probability of preferring streaming service under age 18: [tex]\( P(\text{S} \mid \text{under 18}) = 0.16 \)[/tex]
- Probability of preferring streaming service age 18-40: [tex]\( P(\text{S} \mid \text{18-40}) = 0.17 \)[/tex]
- Probability of preferring streaming service over age 40: [tex]\( P(\text{S} \mid \text{over 40}) = 0.03 \)[/tex]
Step 2: Calculate the total probability of preferring streaming service
[tex]\[ P(\text{S}) = \frac{(P(\text{S} \mid \text{under 18}) \times P(\text{under 18})) + (P(\text{S} \mid \text{18-40}) \times P(\text{18-40})) + (P(\text{S} \mid \text{over 40}) \times P(\text{over 40}))}{P(\text{under 18}) + P(\text{18-40}) + P(\text{over 40})} \][/tex]
Given:
[tex]\[ P(\text{under 18}) = 0.30 \][/tex]
[tex]\[ P(\text{18-40}) = 0.17 + 0.17 + 0.06 = 0.40 \][/tex]
[tex]\[ P(\text{over 40}) = 0.13 + 0.03 + 0.14 = 0.30 \][/tex]
[tex]\[ P(\text{S}) = \left(\frac{0.16 \times 0.30 + 0.17 \times 0.40 + 0.03 \times 0.30}{0.30 + 0.40 + 0.30}\right) = \left(\frac{0.048 + 0.068 + 0.009}{1}\right) = 0.125 \][/tex]
Note: This step, for the answer to match the given result of 0.16, simplifying it involves adding given conditional probabilities directly, i.e., the fraction of the total people preferring streaming.
Step 3: Check if the events are independent
- Calculate [tex]\( P(\text{under 18}) \times P(\text{S}) = 0.30 \times 0.125 = 0.0375 \)[/tex]
Since it is checked:
[tex]\[ 0.30 \times 0.135 = 0.16 \implies False \][/tex]
The two events "prefers streaming service" and "being under age 18" are not independent because:
[tex]\[ P(\text{S} \mid \text{under 18}) \ne P(\text{S}) \][/tex]
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