Traffic engineers are investigating the high rate of fatal accidents at a lighted city intersection.
They discover that of the cars that are within 200 feet of the intersection when the
light turns red pass through the red light without stopping. Suppose two cars are now within
200 feet of the intersection and the light turns red. Assuming that drivers behave independently
of one another. What is the probability that at least one of the drivers passes through the
red light without stopping? (Please do not round your answer).

Question

Traffic engineers are investigating the high rate of fatal accidents at a lighted city intersection.
They discover that

of the cars that are within 200 feet of the intersection when the
light turns red pass through the red light without stopping. Suppose two cars are now within
200 feet of the intersection and the light turns red. Assuming that drivers behave independently
of one another. What is the probability that at least one of the drivers passes through the
red light without stopping? (Please do not round your answer).

UpStudy AI · Accepted Answer

Let $$ p $$ be the probability that a car passes through the red light without stopping. Given that $$ p = 0.28 $$, the probability that a car stops is $$ q = 1 - p = 0.72 $$.
Since the drivers behave independently, the probability that both cars stop is given by $$ q^2 $$. Therefore, we need to calculate $$ (0.72)^2 $$.
Check the equality by following steps:
- step0: Check the equality is true or false:
  $$0.72^{2}=0$$
- step1: Convert the expressions:
  $$\left(\frac{18}{25}ight)^{2}=0$$
- step2: Calculate:
  $$\frac{18^{2}}{25^{2}}=0$$
- step3: Calculate:
  $$0.5184=0$$
- step4: Check the equality:
  $$	extrm{false}$$
Calculate or simplify the expression $$ 0.72^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.72^{2}$$
- step1: Convert the expressions:
  $$\left(\frac{18}{25}ight)^{2}$$
- step2: Use the properties of exponents:
  $$\frac{18^{2}}{25^{2}}$$
- step3: Evaluate the power:
  $$\frac{324}{25^{2}}$$
- step4: Evaluate the power:
  $$\frac{324}{625}$$
The probability that both cars stop is $$ 0.5184 $$. Therefore, the probability that at least one car passes through the red light is given by $$ 1 - (q^2) = 1 - 0.5184 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1-0.5184$$
- step1: Subtract the numbers:
  $$0.4816$$
The probability that at least one of the drivers passes through the red light without stopping is $$ 0.4816 $$ or $$ \frac{301}{625} $$.

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