Question

2. You are designing a puzzle video game that requires fast reaction skills. Note that games such as this can be used in medical settings as methods of psychological and mental assessment, and do not only have uses for entertainment.

You are creating a game where players are supposed to do a simple task, but the simple task changes when the background colour of the game changes, which happens unpredictably. There are four possible background colours, where players are told that each one is associated with a specific described task for them to complete. One round of the game consists of each background colour appearing once in a random order (for example, red, green, blue, yellow). The length of time that a background colour lasts before it changes has an Exponential distribution with parameter λ, but you have not decided on the value of λ yet. If Ti is the length of time that the ith background colour lasts, then Ti ∼ Exp(λ).

(a) Recall that one ‘round’ of the game consists of all four background colours appearing once each. First, define a new random variable for the length of time that a round lasts, and express it in terms of the Ti random variables. Second, state the distribution of the new random variable in terms of λ. Third, suggest a value for λ such that the expected length of time for a single round would be 12 seconds.

(b) If this game was being used in a medical setting to assess mental attentiveness or reaction time, then there might be worry that a single round of the game would be a poor assessment of the desired trait in the patient. Collecting more information would be better, and it might also be better to vary the order that the coloured backgrounds appear. You decide that the game will have multiple rounds, and there will be one round for every single possible order that the four colours can appear in (i.e., there will be one ‘red, green, blue, yellow’ round, one ‘green, red, yellow, blue’ round, etc.). In between each round, there will be 2 seconds of black screen time to offer a moment for the patient to gather themselves. Calculate the expected length of a whole game from start to finish, using your suggested λ from Part (a). Do not include any black screen time before the first round or after the last round.

Answer 1