Mathematics, 06.10.2019 09:02 cpcoolestkid4

Apolice officer is using a radar device to check motorists’ speeds. prior to beginning the speed check, the officer estimates that 40 percent of motorists will be driving more than 5 miles per hour over the speed limit. assuming that the police officer’s estimate is correct, what is the probability that among 4 randomly selected motorists, the officer will find at least 1 motorist driving more than 5 miles per hour over the speed limit?

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Answer from: Yeetious

The probability is 0.8704

Step-by-step explanation:

A Binomial distribution apply when we have n identical and independent events with two possible results, success or fail, and a probability p of success and 1-p of fail. Then, the probability that from the n events, x are success is:

$P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}$

Then, the number of motorist driving more than 5 miles per hour over the speed limit follow a Binomial Distributions, because we have 4 randomly selected motorists with two possible options: they are driving more than 5 miles per hour over the speed limit or they aren't, and the probability that they are driving more that 5 miles per hour over the speed is 0.4.

Finally, the probability P that among 4 randomly selected motorists, the officer will find at least 1 motorist driving more than 5 miles per hour over the speed limit is:

P = P(1) + P(2) + P(3) + P(4)

Where n is equal to 4 and p is equal to 0.4. Replacing the values for every x, we get:

$P(1)=\frac{4!}{1!(4-1)!}*0.4^{1}*(1-0.4)^{4-1}=0.3456$

$P(2)=\frac{4!}{2!(4-2)!}*0.4^{2}*(1-0.4)^{4-2}=0.3456$

$P(3)=\frac{4!}{3!(4-3)!}*0.4^{3}*(1-0.4)^{4-3}=0.1536$

$P(4)=\frac{4!}{4!(4-4)!}*0.4^{4}*(1-0.4)^{4-4}=0.0256$

Finally, P is equal to:

P = 0.3456 + 0.3456 + 0.1536 + 0.0256

P = 0.8704

Answer from: NeonPlaySword

The probability that among 4 randomly selected motorists, the officer will find at least 1 motorist driving more than 5 miles per hour over the speed limit is 0.8704.

Given :

Prior to beginning the speed check, the officer estimates that 40 percent of motorists will be driving more than 5 miles per hour over the speed limit.

According to the Binomial distribution the probability is given by :

$\rm P(x) = \dfrac{n!}{x!(n-x)!}\times p^x \times (1-p)^{n-x}$

where n is the event and x is the success.

The probability that among 4 randomly selected motorists, the officer will find at least 1 motorist driving more than 5 miles per hour over the speed limit is:

$\rm P = P(1) + P(2) + P(3) + P(4)$

Here, n = 4 and P = 0.4.

$\rm P(1)= \dfrac{4!}{1!(4-1)!}\times 0.4^1 \times (1-0.4)^{4-1} = 0.3456$

$\rm P(2)= \dfrac{4!}{2!(4-2)!}\times 0.4^2 \times (1-0.4)^{4-2} = 0.3456$

$\rm P(3)= \dfrac{4!}{3!(4-3)!}\times 0.4^3 \times (1-0.4)^{4-3} = 0.1536$

$\rm P(4)= \dfrac{4!}{1!(4-4)!}\times 0.4^4 \times (1-0.4)^{4-4} = 0.0256$

Now, the value of P is given by:

$\rm P=P(1)+P(2)+P(3)+P(4)$

P = 0.3456 + 0.3456 + 0.1536 + 0.0256

P = 0.8704

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Answer from: Quest

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Answer from: Quest

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