Problem
Assume a predictive model can tell from a user's Twitter posts whether the user is a male or a female.
- 20% of the users are females.
- If a person is a female, the model is 90% inclined to answer correctly.
- If a person is a male, the model is 70% inclined to answer correctly.
Assume the model describes the user being a female. What is the probability that the user is actually a male?
Attempt
Applying Bayes' Theorem, I created the tree and from this formulated the following equation.
$$ P(male | wrong) = \frac{P(male)P(wrong | male)}{P(wrong)} $$ $$ ... =\frac{P(male)P(wrong | male)}{P(male)P(wrong|male)+P(female)P(wrong|female)}$$
Plugging into this formula, I get $$ P(male | wrong) = \frac{0.8 \times 0.3}{0.8 \times 0.3+0.2 \times 0.1}=.93 \implies 93\%$$
Notes
Is my approach to this problem correct? I assume if it is, then the calculation should be too. Otherwise, any guidance on how to formulate Bayes' theorem in this case will be greatly appreciated!
Take $ wrong_{f} $ to be the probability that the machine is wrong in saying the individual is female (when he is a male).
$$ P(male | wrong_{f}) = \frac{P(male)P(wrong_{f} | male)}{P(wrong_{f})} $$ $$ ... =\frac{P(male)P(wrong_{f} | male)}{P(male)P(wrong_{f}|male)+P(female)P(wrong_{m}|female)}$$
Plugging in, $$ P(male | wrong_{f}) = \frac{0.8 \times 0.3}{0.8 \times 0.3+0.2 \times 0.9}=.57 \implies 57\%$$