Environmental Decisionmaking

Problem Set 9: Intro to Probability

1) For the following problems, assume that you have a fair coin (i.e. there is an equal probability of tossing a head or a tail).

a) (1 pt) If you flip the coin twice, what is the probability of getting heads both times?

The probability of getting heads in a single roll is 0.5. To get heads both times you have to get heads the first time and the second, so we need to multiply the probabilities:

0.5 * 0.5 = 0.25

b) (1 pts) What is the probability of getting one head and one tail? (order isn't important)

There are two ways this can happen: either you can get heads the first time and tails the second (with a probability of 0.5 * 0.5 = 0.25), or you can get tails the first time and heads the second (which also has a probability of 0.25). Since you can get the given outcome with either toss, the probability is:

0.25 + 0.25 = 0.5

c) (2 pts) If you toss the coin three times, what is the probability of getting tails all three times?

0.5 * 0.5 * 0.5 = 0.125 (or 1/8, which makes sense)

d) (2 pts) What is the probability of getting one head and two tails (order isn't important).

You can get that one of three ways: HTT, THT, TTH, each of which has a probability of 0.125. Hence, the total probability is 3 * 0.125 = 0.375 (or 3/8).

2) The fraction of people in a population who have a certain disease is 0.01. A diagnostic test is available to test for the disease. For a healthy person, the chance of being falsely diagnosed as having the disease is 0.05, while for someone with the disease the chance of being falsely diagnosed as healthy is 0.2. Suppose the test is performed on a person selected at random from the population.

a) (2 pts) What is the probability that the test shows a positive result (meaning that the person is diagnosed as diseased, perhaps correctly, perhaps not)?

There are two ways this can happen. First, you might not have the disease (P(ND)=0.99). If that is the case, then the chance of the test being wrong and showing a positive result, P(+|ND), is 0.05. Hence, the probability of both of these being the case is 0.99 * 0.05 = 0.0495. You also might have the disease, P(D)=0.01. If that is the case, then the probability of getting a true positive P(+|D) is 0.8. Hence, this probability is: 0.01 * 0.8 = 0.008. So, the total probability of getting a positive result is P(+) = 0.0495 + 0.008 = 0.0575, or 5.75%.

b) (2 pts) What is the probability that the person selected at random is one who has the disease, but is diagnosed as healthy?

P(- and disease) = P(D)*P(-|D) = 0.01 * 0.2 = 0.002.

c) (2 pts) Suppose that the test shows a positive result. What is the probability that the person actually has the disease?

Here we can use what we did in the first part. P(+ and D) = 0.008. P(+ and ND) = 0.0495. We saw above that 5.75% of people are diagnosed as positive, and 0.8% of people are correctly diagnosed as positive, so:

P(D|+) = 0.008/0.0575 = 0.13913.

So, if you are diagnosed as positive, you have about a 14% chance of actually having the disease; significantly more than the 1% chance you had before the new information, but certainly not that 80% chance that many people would have thought.

3) (4 pts) Assume you are a decision-maker. One scientist tells you that they are 90% confident that there is oil to be found in a fragile wilderness. Another tells you that the probability that they can set up the equipment to drill for oil and not cause irreperable harm to the ecosystem is 85%. A third assures you that they can build a safe pipeline. The only time the pipeline wouldn't be safe would be if there was an earthquake, but there is a 90% chance that no earthquake will occur over the entire period of operations, though if there was an earthquake, the rupture would be quite damaging to the ecosystem. Finally, some activists have claimed that the mere existence of the pipeline would disrupt migration patterns, which would cause cascading effects throughout the ecosystem and lead to the loss of ecosystem function, but the industry officials assure you that their studies show that there is an 80% probability that the migratory animals will find another route. Your colleagues are comforted that there seems to be about an 80% probability that everything will be fine, and suggest that you support the project. You object, and say that the probability of finding oil and not harming the ecosystem is much lower, even according to the industry sources themselves. About what is the probability that oil will be found and the ecosystem will come to no harm, using the numbers given?

For there to be no problem, all of these have to happen at the same time. That probability is easy to figure:

There is a 45% chance that significant harm will come to the ecosystem, which might result in a very different policy!

See reading.