Question

The developers of a new online game have determined from preliminary testing that the scores of players on the first level of the game can be modelled satisfactorily by a Normal distribution with a mean of 185 points and a standard deviation of 28 points. They would like to vary the difficulty of the second level in this game, depending on the player’s score in the first level. (a) The developers have decided to provide different versions of the second level for each of the following groups: (i) those whose score on the first level is in the lowest 25% of scores ii) those whose score on the first level is in the middle 50% of scores (iii) those whose score on the first level is in the highest 25% of scores. Use the information given above to determine the cut-off scores for these groups. (You may round each of your answers to the nearest whole number.) (b) In the second level of the game, the developers have also decided to give players an opportunity to qualify for a bonus round. Their stated aim is that players from group (i) should have 75% chance of qualifying for the bonus round, players from group (ii) should have 55% chance of qualifying for the bonus round and that players from group (iii) should have 30% chance of qualifying for this round. Let ?, ? and ? respectively denote the events that a player’s score on the first level was in the lowest 25% of scores, the middle 50% of scores and the highest 25% of scores, and let ? denote the event that the player qualifies for the bonus round. Use event notation to express the developers’ aim as a set of conditional probabilities. (c) Based on the developers’ stated aim, find the total probability that a randomly chosen player will qualify for the bonus round. (d) Given that a player has qualified for the bonus round, what is the probability that the player’s score on the first level was in the middle 50% of scores for that level? (e) Given that a player has not qualified for the bonus round, what is the probability that the player’s score on the first level was in the lowest 25% of scores for that level?

Answer 1