Automobiles are testing for emissions. It is determined that 30 % of automobiles whose emissions levels are inspected fail the inspection. We shall conduct the experiment of inspecting car emissions upon individual cars. a.) What random variable computes the probability that r cars are inspected for their emission levels until one car fails? b.) Write down the probability mass function for your answer in a.) c.) What is the probability that 5 cars are inspected until a car fails its emissions test?
Automobiles are testing for emissions. It is determined that 30 % of automobiles whose emissions levels are inspected fail the inspection. We shall conduct the experiment of inspecting car emissions upon individual cars. a.) What random variable computes the probability that r cars are inspected for their emission levels until one car fails? b.) Write down the probability mass function for your answer in a.) c.) What is the probability that 5 cars are inspected until a car fails its emissions test?
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![### Probability and Emissions Inspection
Automobiles are tested for emissions. It is determined that **30% of automobiles whose emissions levels are inspected fail the inspection**. We shall conduct the experiment of inspecting car emissions on individual cars.
#### Questions:
**a.) What random variable computes the probability that \(x\) cars are inspected for their emission levels until one car fails?**
In this scenario, we are dealing with a geometric distribution. The geometric random variable \(X\) represents the number of trials (car inspections) needed to get the first success (failure in emission test).
**b.) Write down the probability mass function for your answer in a.)**
The probability mass function (PMF) for a geometric random variable \(X\), where \( X = x \) is given by:
\[ P(X = x) = (1-p)^{x-1}p \]
In our case, \( p = 0.3 \) (the probability of an automobile failing the emission test). Therefore, the PMF is:
\[ P(X = x) = (1-0.3)^{x-1} \cdot 0.3 \]
\[ P(X = x) = (0.7)^{x-1} \cdot 0.3 \]
**c.) What is the probability that 5 cars are inspected until a car fails its emissions test?**
To find this probability, substitute \( x = 5 \) into the PMF formula:
\[ P(X = 5) = (0.7)^{5-1} \cdot 0.3 \]
\[ P(X = 5) = (0.7)^{4} \cdot 0.3 \]
\[ P(X = 5) = 0.2401 \cdot 0.3 \]
\[ P(X = 5) = 0.07203 \]
Therefore, the probability that 5 cars are inspected until one car fails the emissions test is \(0.07203\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45fdd75f-924d-4ac4-9d15-03fc1f26bc02%2F77d76aaf-9a28-4f37-a107-87109039f877%2Fe2nnn8_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability and Emissions Inspection
Automobiles are tested for emissions. It is determined that **30% of automobiles whose emissions levels are inspected fail the inspection**. We shall conduct the experiment of inspecting car emissions on individual cars.
#### Questions:
**a.) What random variable computes the probability that \(x\) cars are inspected for their emission levels until one car fails?**
In this scenario, we are dealing with a geometric distribution. The geometric random variable \(X\) represents the number of trials (car inspections) needed to get the first success (failure in emission test).
**b.) Write down the probability mass function for your answer in a.)**
The probability mass function (PMF) for a geometric random variable \(X\), where \( X = x \) is given by:
\[ P(X = x) = (1-p)^{x-1}p \]
In our case, \( p = 0.3 \) (the probability of an automobile failing the emission test). Therefore, the PMF is:
\[ P(X = x) = (1-0.3)^{x-1} \cdot 0.3 \]
\[ P(X = x) = (0.7)^{x-1} \cdot 0.3 \]
**c.) What is the probability that 5 cars are inspected until a car fails its emissions test?**
To find this probability, substitute \( x = 5 \) into the PMF formula:
\[ P(X = 5) = (0.7)^{5-1} \cdot 0.3 \]
\[ P(X = 5) = (0.7)^{4} \cdot 0.3 \]
\[ P(X = 5) = 0.2401 \cdot 0.3 \]
\[ P(X = 5) = 0.07203 \]
Therefore, the probability that 5 cars are inspected until one car fails the emissions test is \(0.07203\).
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