Final answer:
The ventromedial prefrontal cortex within the prefrontal cortex is responsible for encoding the utility of a gamble and shows increased activity related to the probability of accepting a gamble.
Step-by-step explanation:
The brain regions that encode the utility of a gamble and exhibit an increased activity that is linearly related to the probability of accepting a gamble are located in the prefrontal cortex. Specifically, the ventromedial prefrontal cortex (vmPFC) is involved in processing risk and reward and is responsible for determining the value of a gamble. The vmPFC, along with the medial temporal lobe, is involved in various executive functions including decision-making, working memory, and inhibiting distractions that enable focused goal-directed behavior. These cognitive functions are essential when evaluating decisions like whether to accept a gamble. The other options given, such as the motor cortex, visual cortex, and auditory cortex, are associated with different regions of the brain responsible for processing movement, visual information, and auditory information, respectively, and are not directly related to the encoding of the utility of a gamble.