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stiv31 [10]
3 years ago
12

On average, there are 12 potholes per mile on a particular stretch of the state highway. Suppose the potholes follow a Poisson d

istribution on the highway. a. Find the probability of finding fewer than two potholes in a quarter-mile stretch of the highway. (Do not round intermediate calculations. Round your final answer to 4 decimal places.) b. Find the probability of finding more than one pothole in a quarter-mile stretch of the highway. (Do not round intermediate calculations. Round your final answer to 4 decimal places.)
Mathematics
1 answer:
Lunna [17]3 years ago
5 0

Answer:

a) 0.1992 = 19.92% probability of finding fewer than two potholes in a quarter-mile stretch of the highway.

b) 0.8008 = 80.08% probability of finding more than one pothole in a quarter-mile stretch of the highway.

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

In which

x is the number of sucesses

e = 2.71828 is the Euler number

\mu is the mean in the given interval.

a. Find the probability of finding fewer than two potholes in a quarter-mile stretch of the highway.

Mean of 12 potholes per mile, which means that in a quarter-mile stretch, the mean is \mu = \frac{12}{4} = 3

This probability is:

P(X < 2) = P(X = 0) + P(X = 1)

In which

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

P(X = 0) = \frac{e^{-3}*(3)^{0}}{(0)!} = 0.0498

P(X = 1) = \frac{e^{-3}*(3)^{1}}{(1)!} = 0.1494

P(X < 2) = P(X = 0) + P(X = 1) = 0.0498 + 0.1494 = 0.1992

0.1992 = 19.92% probability of finding fewer than two potholes in a quarter-mile stretch of the highway.

b. Find the probability of finding more than one pothole in a quarter-mile stretch of the highway.

This is

P(X > 1) = 1 - P(X < 2)

We have that P(X < 2) = 0.1992

So

P(X > 1) = 1 - P(X < 2) = 1 - 0.1992 = 0.8008

0.8008 = 80.08% probability of finding more than one pothole in a quarter-mile stretch of the highway.

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