10   4 Shai has created a video game. She makes her game free to play, with players able to buy gems to spend in the game for bonuses. She creates a model to predict the income that she can expect from her game. Shai’s data shows that the majority of players do not spend any money on gems. A small number of players, known as ‘whales’, make up the majority of the revenue for games by buying the gems. [Source: https://deltadna.com/blog/how-whales-spend/] (a) Give a reason why the normal distribution is unlikely to be a suitable model for the amount a randomly selected player spends in a given month. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Shai instead chooses to model the spend, in pounds, of a randomly selected whale in a given month. She uses the distribution W ~ N(50, 100) She assumes that the amount a whale spends each month is independent from their spend in previous months. (b) Find  P(W < 40) (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) Using Shai’s model, calculate the probability that a randomly selected whale has a mean spend of less than £40 a month over a year. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (d) Explain why the probability calculated in part (c) must be smaller than the probability calculated in part (b). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 11   Question 4 continued (e) Explain why  P(W  40)  and  P(W < 40)  are equal. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (f) Calculate the probability that a randomly selected whale spends more than £40 in each of four consecutive months. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (g) Calculate the probability that a randomly selected whale spends more than £40 in at least 8 of the months in a one‑year period. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (h) Calculate the probability that a randomly selected whale spends less than £40 in a month that they spent more than £30 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________