Question

In an office there are two boxes of thumb drives: Box \(A_{1}\) contains seven 100 GB drives and three 500 GB drives, and box \(A_{2}\) contains two 100 GB drives and eight 500 GB drives. A person is handed a box at random with prior probabilities \(P\left(A_{1}\right)=\frac{2}{3}\) and \(P\left(A_{2}\right)=\frac{1}{3}\), possibly due to the boxes' respective locations. A drive is then selected at random and the event \(B\) occurs if it is a \(500 \mathrm{~GB}\) drive. Using an equally likely assumption for each drive in the selected box, compute \(P\left(A_{1} \mid B\right)\) and \(P\left(A_{2} \mid B\right)\)

Short answer

\(P\left(A_{1} \mid B\right) = \frac{3}{7}\) and \(P\left(A_{2} \mid B\right) = \frac{4}{7}\)

Step-by-step solution

Calculate prior probabilities

The prior probabilities are given in the problem statement as \(P\left(A_{1}\right)=\frac{2}{3}\) and \(P\left(A_{2}\right)=\frac{1}{3}\)

Calculate conditional probabilities

We are given that there is an equal probability of selecting any drive within a box. In box \(A_{1}\right), three out of ten drives are 500 GB drives. So \(P\left(B|A_{1}\right)=\frac{3}{10}\). In box \(A_{2}\right), eight out of ten drives are 500 GB drives. So \(P\left(B|A_{2}\right)=\frac{8}{10}\)

Calculate total probability of B

We use the law of total probability to calculate the total probability of selecting a 500 GB thumb drive: \(P\left(B\right)= P\left(A_{1}\right) * P\left(B|A_{1}\right) + P\left(A_{2}\right) * P\left(B|A_{2}\right) = \frac{2}{3} * \frac{3}{10} + \frac{1}{3} * \frac{8}{10} = \frac{6}{30} + \frac{8}{30} = \frac{14}{30}\)

Apply Bayes' theorem

We can now apply Bayes' theorem to calculate the probability of having come from each box given that a 500 GB drive was selected. \(P\left(A_{1}| B\right)= \frac{P\left(B|A_{1}\right) * P\left(A_{1}\right)}{P\left(B\right)} = \frac{\frac{3}{10} * \frac{2}{3}}{\frac{14}{30}} = \frac{3}{7}\). Similarly, \(P\left(A_{2}| B\right)= \frac{P\left(B|A_{2}\right) * P\left(A_{2}\right)}{P\left(B\right)} = \frac{\frac{8}{10} * \frac{1}{3}}{\frac{14}{30}} = \frac{4}{7}\)