Probability of an event basedon given variables.

[FrankJ777]

For some time now I've been trying to figure out probably for a problem of the following form.

Say a criminal profiler is trying to determine the probability that someone is a criminal based on statistical information.

60% of people who have mustaches are criminals.
70% of people who wear black hats are criminals.
80% of people who wear sunglasses are criminals.

If the profiler is profiling an individual who wears sunglasses, a black hat, and has a mustache; how would she determine the probability that this individual is a criminal?

By the way, this is not a homework problem, I'm just trying to understand how to apply probability better.
Thanks a lot.

 

[ImaLooser]

For some time now I've been trying to figure out probably for a problem of the following form.

Say a criminal profiler is trying to determine the probability that someone is a criminal based on statistical information.

60% of people who have mustaches are criminals.
70% of people who wear black hats are criminals.
80% of people who wear sunglasses are criminals.

If the profiler is profiling an individual who wears sunglasses, a black hat, and has a mustache; how would she determine the probability that this individual is a criminal?

By the way, this is not a homework problem, I'm just trying to understand how to apply probability better.
Thanks a lot.

The events may or may not be independent. If they aren't independent, then there is no answer other than collecting data and measuring it directly.

 

[FrankJ777]

Could one at least establish an upper or lower probability limit? Would the individual in the example have at least a 80% probability of being a criminal based on the criteria that 80% of people who wear sunglasses are criminals? On the other would the probability of the individual being a criminal be at least 60% based on the "fact" that 60% of people with mustaches are criminals?

 

[mathman]

80% would be lower limit.
If the criteria are independent, then the probability of being a criminal is
1 - prob(at least one does not hold) = 1 - .4x.3x.2 = 97.6%.

 

[blue_raver22]

I would approach this problem by using conditional probability. In this case, we are looking to determine the probability that an individual is a criminal given that they wear sunglasses, a black hat, and have a mustache. This can be represented as P(criminal|sunglasses, black hat, mustache).

Using Bayes' Theorem, we can calculate this probability by multiplying the individual probabilities of each variable (sunglasses, black hat, mustache) and then dividing by the overall probability of wearing sunglasses, a black hat, and having a mustache. This can be represented as follows:

P(criminal|sunglasses, black hat, mustache) = (P(sunglasses|criminal) * P(black hat|criminal) * P(mustache|criminal)) / P(sunglasses, black hat, mustache)

Plugging in the given information, we get:

P(criminal|sunglasses, black hat, mustache) = (0.80 * 0.70 * 0.60) / (P(sunglasses, black hat, mustache))

Now, the overall probability of wearing sunglasses, a black hat, and having a mustache can be calculated by adding the probabilities of each individual combination that meets this criteria. For example, someone who wears sunglasses and a black hat but doesn't have a mustache, or someone who wears a black hat and has a mustache but no sunglasses. This can be represented as follows:

P(sunglasses, black hat, mustache) = P(sunglasses, black hat, mustache, criminal) + P(sunglasses, black hat, mustache, not criminal)

We already have the probabilities for P(sunglasses, black hat, mustache, criminal) from the given information. To calculate P(sunglasses, black hat, mustache, not criminal), we can use the complement rule: P(not criminal) = 1 - P(criminal).

Once we have the overall probability of wearing sunglasses, a black hat, and having a mustache, we can plug it back into our initial equation to calculate the probability of an individual being a criminal given that they wear sunglasses, a black hat, and have a mustache.

I hope this helps you better understand how to apply probability in this type of problem. It's important to note that these probabilities are based on statistical information and may not accurately represent an individual's