The developers of a new online game have determined frompreliminary testing that the scores of players on the first levelof the game can be modelled satisfactorily by a Normal distributionwith a mean of 185 points and a standard deviation of 28 points.They would like to vary the difficulty of the second level in thisgame, depending on the player’s score in the first level.
(a) The developers have decided to provide different versions ofthe second level for each of the following groups:
(i) those whose score on the first level is in the lowest 25% ofscores
ii) those whose score on the first level is in the middle 50% ofscores
(iii) those whose score on the first level is in the highest 25%of scores. Use the information given above to determine the cut-offscores for these groups. (You may round each of your answers to thenearest whole number.)
(b) In the second level of the game, the developers have alsodecided to give players an opportunity to qualify for a bonusround. Their stated aim is that players from group (i) should have75% chance of qualifying for the bonus round, players from group(ii) should have 55% chance of qualifying for the bonus round andthat players from group (iii) should have 30% chance of qualifyingfor this round. Let ?, ? and ? respectively denote the events thata player’s score on the first level was in the lowest 25% ofscores, the middle 50% of scores and the highest 25% of scores, andlet ? denote the event that the player qualifies for the bonusround. Use event notation to express the developers’ aim as a setof conditional probabilities.
(c) Based on the developers’ stated aim, find the totalprobability that a randomly chosen player will qualify for thebonus round.
(d) Given that a player has qualified for the bonus round, whatis the probability that the player’s score on the first level wasin 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 levelwas in the lowest 25% of scores for that level?
