Where's my Book?

Solution

Answer: 2/3

Let the possible locations of the textbook be \[ T = \text{Town Square}, \quad H = \text{Home}, \quad O = \text{Other location}. \] Jason believes there is an \(80\%\) chance the book is either at Town Square or at home. He assigns \[ P(T) = 0.4, \quad P(H) = 0.4. \] Therefore the remaining probability is \[ P(O) = 1 - 0.8 = 0.2. \] We are told that the book is not at home. Let this information be the event \[ E = H^c. \] So the remaining possible locations are Town Square or another location. The conditional probability that the book is at Town Square given that it is not at home is \[ P(T \mid H^c) = \frac{P(T \cap H^c)}{P(H^c)}. \] Since Town Square and Home cannot both occur, \[ P(T \cap H^c) = P(T) = 0.4. \] Also \[ P(H^c) = 1 - P(H) = 1 - 0.4 = 0.6. \] Thus \[ P(T \mid H^c) = \frac{0.4}{0.6}. \] \[ P(T \mid H^c) \approx 0.667. \]