# HELP PLEASE Probability of Waste dump sites

• Fear_of_Math
In summary, the conversation discusses a problem involving a federal agency trying to determine the probability of federal law violations in two waste dump projects. The administrator believes the violations are disjoint but actually they are independent. The conversation also includes questions and confusion about the difference between disjoint and independent events.

#### Fear_of_Math

HELP PLEASE! Probability of Waste dump sites

Okay, here's my problem:
A federal agency is deciding wqhich of two waste dump projects to inviestigate. A top administrator estimates that the probability of federal law violations is 0.30 in the first site and 0.25 at the second project. He also believes the occurences of violations in these two projects are disjoint.

#1. what is the probability of federal law violations in the first or second project
So I'm thinking it's just p(AnB)=0.30-0.25, but I'm not sure. My prof has some examples and horrible explanaitions that don't give just probability, so I'm lost here. I also thought it could be p(AuB)= 0.3+0.25, but I am so lost as stats and math are not my strong point.

#2. Given that there is not a federal law violation in the first project, find the probability that there is a federal law violation in the second project.
Is this supposed to be p(B)=0.25? I know that's just the easy answer, but really, if there is non happeing in the first project, that means p(Abar) =0.70, so would it be 0.70+0.25? or 0.70-0.25? Once again, I have no idea

#3. In reality, the administrator confused disjoint and independent, and the events are actually independent. Anser #1 and #2 with the correct information.
Okay, so dijoint means they just don't happen at the same time, and independent means ...I don't know, how would all this change?

Any help is appreciated. Explanaitions really work wonders, especially if you can show me with those venn diagrams. I'm just so very very lost in this class they force me to take for my degree. Kudos for those of you who rock this stuff =)

Let $$A$$ be the event that the first dump commits a violation.

Let $$B$$ be the event that the second dump commits a violation.

Disjoint means $$P(A \cap B) = 0$$

It also means $$P(A \cap \overline{B}) = P(A)$$ and $$P( \overline{A} \cap {B}) = P(B)$$

Independence means $$P (A \cap B) = P(A)P(B)$$

And if $$A$$ and $$B$$ are independent, so are $$A$$ and $$\overline{B}$$, $$\overline{A}$$ and $$B$$, and $$\overline{A}$$ and $$\overline{B}$$

1) $$P(A \cup B) = P(A) + P (B) - P(A \cap B)$$

2) $$P(B| \overline{A}) = \frac {P(B \cap \overline{A})}{P(\overline{A})} = \frac {P(B) + P( \overline{A}) - P(B \cap \overline{A})}{P(\overline{A})}$$

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Fear_of_Math said:
Okay, here's my problem:
A federal agency is deciding wqhich of two waste dump projects to inviestigate. A top administrator estimates that the probability of federal law violations is 0.30 in the first site and 0.25 at the second project. He also believes the occurences of violations in these two projects are disjoint.
"Disjoint" in this situation would mean there cannot be violations in BOTH sites- one excludes the other.
Do you mean "independent" rather than "disjoint" here?

#1. what is the probability of federal law violations in the first or second project
So I'm thinking it's just p(AnB)=0.30-0.25, but I'm not sure.
First AnB is "A and B", not "A or B". Second p(AnB)= 0 if A and B are "disjoint" (mutually exclusive) or p(A)p(B), if they are independent, not P(A)- P(B).

P(A or B)= p(A)+ p(B)- p(AnB) and so either .30+ .25= .55 if they are mutually exclusive, .30+ .25- (.30)(.25)= .55-.075= .475 if they are independent.

My prof has some examples and horrible explanaitions that don't give just probability, so I'm lost here. I also thought it could be p(AuB)= 0.3+0.25, but I am so lost as stats and math are not my strong point.

#2. Given that there is not a federal law violation in the first project, find the probability that there is a federal law violation in the second project.
If they are "disjoint" (mutually exclusive), then p(B|A) (probability of B given that A is true)= 0, by definition of "mutually exclusive". If independent, then p(B)= .25 irrelevant of A.

Is this supposed to be p(B)=0.25? I know that's just the easy answer, but really, if there is non happeing in the first project, that means p(Abar) =0.70, so would it be 0.70+0.25? or 0.70-0.25? Once again, I have no idea

#3. In reality, the administrator confused disjoint and independent, and the events are actually independent. Anser #1 and #2 with the correct information.
Okay, so dijoint means they just don't happen at the same time, and independent means ...I don't know, how would all this change?
I did not notice until after my previous answers the whole problem is about a confusion of "disjoint" and "independent" so I have alread answered this!

Any help is appreciated. Explanaitions really work wonders, especially if you can show me with those venn diagrams. I'm just so very very lost in this class they force me to take for my degree. Kudos for those of you who rock this stuff =)

Thank you so much for putting this in perspective for me. I have a better understanding now then I did those three days in the lecture!

## 1. What is the probability of a waste dump site being located near residential areas?

The probability of a waste dump site being located near residential areas depends on various factors such as zoning laws, environmental regulations, and community resistance. In some cases, waste dump sites may be intentionally placed near residential areas due to political or economic reasons. However, in most cases, the probability is low as authorities try to minimize the potential health and environmental risks for nearby residents.

## 2. How do scientists calculate the probability of a waste dump site causing environmental pollution?

Scientists use various statistical and modeling techniques to calculate the probability of a waste dump site causing environmental pollution. This includes evaluating the types and quantities of waste being dumped, the location and characteristics of the site, and the potential pathways for pollution to spread. Environmental monitoring and risk assessment are also important tools in determining the probability of pollution.

## 3. Can the probability of a waste dump site causing health hazards be reduced?

Yes, the probability of a waste dump site causing health hazards can be reduced through proper waste management practices and regulations. This includes implementing measures to prevent leaks and spills, proper disposal of hazardous waste, and regular monitoring of air, water, and soil quality. Community involvement and awareness can also play a crucial role in reducing the probability of health hazards.

## 4. What are the long-term effects of waste dump sites on the environment?

The long-term effects of waste dump sites on the environment depend on various factors such as the type of waste, the location and size of the site, and the effectiveness of waste management practices. These effects can include contamination of air, water, and soil, loss of biodiversity, and potential health hazards for humans and wildlife. It is important to properly monitor and manage waste dump sites to minimize these long-term effects.

## 5. How can we determine the probability of a waste dump site being safe for human and environmental health?

Determining the probability of a waste dump site being safe for human and environmental health requires a comprehensive assessment of various factors. This includes evaluating the types and quantities of waste being dumped, the location and characteristics of the site, and the potential pathways for pollution to spread. Environmental monitoring, risk assessment, and community involvement are also crucial in determining the safety of a waste dump site.