Apply your knowledge of conditional probability and Bayes’ rule to solve the fol
ID: 3042437 • Letter: A
Question
Apply your knowledge of conditional probability and Bayes’ rule to solve the following problem.
Your company provides consulting services regarding fossil fuels and alternative energy sources. One of the services you offer for a specific region of southern Alaska is an above-ground test (which requires no drilling) that tries to ascertain whether natural gas is present. Past drilling surveys in the region have shown that in 72 instances out of 120, gas was present. This shows the high potential for this region.
Your biggest concern is the “angry phone call” in which the sensors indicate gas and there turns out to be none: developers become furious when they drill based on your positive sensor and then find no gas. The case in which the sensors indicate no gas is of far less concern, because if the developers go with your advice not to drill, they will never know if there was gas or not – and if they drilled against your advice either they will be happy to find gas or they will find out your sensor was correct.
You have an option to invest more engineering in the ground-based sensing technology to improve the likelihood that it detects gas when gas is actually present – we call this the gas detection rate (GDR), currently at 70%. When gas is not present, the sensors correctly state “no gas” 80% of the time.
If 100 customers were to apply the sensors, how many angry phone calls should you expect? (Again, this is the case when the sensor said there is gas but there turns out to be none.) Can you reduce the number of angry phone calls by increasing the GDR? Explain.
Suppose you are a developer and the sensor flag indicates that there is gas present. What is the probability that this turns out to be wrong? Can you reduce this by changing the GDR? Explain.
Your sensors are reengineered to indicate high quality gas, low quality gas, or no gas at all. It turns out that 1/3 of the gas in the region is ‘low quality’ meaning that it is poor quality and costly to extract. Since the (unconditional) probability of gas being present is 0.6 (that is, 72/120), this means that the (unconditional) probability of there being high quality gas is 0.4 and the probability of there being low quality gas is 0.2. When high quality gas is present, the sensors detect it 50% of the time, but indicate low quality 20% of the time. Similarly, when low quality gas is present, the sensors indicate so 50% of the time, but indicate high quality 20% of the time. When there is no gas, the sensors indicate so 80% of the time and are equally likely to indicate high quality gas or low quality gas.
How often would we expect the sensors to detect high quality gas?
When the sensors claim there is high quality gas, what are the chances that it is correct?
Explanation / Answer
Given:
Probability of gas being present in the area = 72/120 = 0.6
Probability of gas being absent in the area = 1 - 0.6 = 0.4
The GDR = 70%, that is, it correctly detects the presence of gas 70% of the times and it correctly detects the absence of gas 80% of the times.
Solution:
Probability of getting angry phone call = Probability of wrong detection of presence of gas
= 1 - 0.7
= 0.3
Expected number of angry phone call = 100 * 0.3 = 30
Yes, angry phone calls can be reduced by increasing the GDR as the detection rate would increase with the use of more sensors.
Given:
The probability of there being high quality gas = 0.4
The probability of there being low quality gas = 0.2
Probability that sensor detect high quality gas when it is present = 0.5
Probability that sensor detect low quality gas when high quality gas is present = 0.2
Probability that sensor detect low quality gas when it is present = 0.5
Probability that sensor detect high quality gas when low quality gas is present = 0.2
Probability that sensor correctly detects that there is no gas present = 0.8
Probability that sensor indicated high quality gas when no gas is present = 0.5
Probability that sensor indicated low quality gas when no gas is present = 0.5
Solution:
The probability of high quality gas being present in the region is 0.4.
Therefore, 40% of the times we can expect the sensor to detect high quality gas.
The probability that sensor detect high quality gas when it is present = 0.5
Therefore, there are 50% chances that it is correct.
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