Question

# You have high standards with respect to truth in advertising, particularly when it comes to hair color. One day at the laundromat, you meet an attractive blonde stranger named Chris and wonder if you should pursue relationship. Unfortunately, you have the nagging belief that Chris golden locks may have been the result of peroxide-presenting the specter of a dark (haired) future. However, you also know several facts about the incidence of dyed hair about your ability to detect fraudulent follicles. You know that $90 \%$ of blonde people in the world are naturally blonde. You have done a personal survey and learned that you are $80 \%$ accurate in your ability to correctly categorize fake hair color as fake and real hair color as real. What is the probability that Chris's hair is fair and that your bleached beliefs were incorrect? Given these facts, should you pursue your relationship with Chris?

Solution

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There are $90\%$ natural blonds among all blonds. We can correctly guess natural hair as natural, and fake hair as fake, $80\%$ of times. We are suspicious in two cases; First, if the blond person is a natural blond ($0.9$ chance) chance, we have a $1-0.8=0.2$ chance of guessing incorrectly that the hair is natural when it is in fact fake. Second, if the blond person is not a natural blond ($1-0.9=0.1$ chance), there is a $0.8$ chance of us guessing that the hair is fake.

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