2025 MI P2 Q9

In a particular stage of a computer game, a player is given a first set of 20 tasks

• If he completes fewer than \(13\) tasks, he has to retry the stage.
• If he completes more than \(16\) tasks, he progresses directly to the next stage.
• Otherwise, he is given a second set of \(10\) tasks.
  • If he completes more than \(7\) of these \(10\) tasks, he progresses to the next stage.
  • Otherwise, he has to retry the stage.

It is known that the player's performances in the two sets of tasks are independent. Let \(X\) and \(Y\) be the number of tasks completed in the first and second sets of tasks respectively.

1. State two assumptions needed for \(X\) to be well-modelled by a binomial distribution.[2]

Assume now that \(X\) has the distribution \(\mathrm{B}(20, 0.7)\) and \(Y\) has the distribution \(\mathrm{B}(10, 0.8)\).

2. Find the probability that the player is given a second set of tasks.[2]
3. Using a tree diagram or otherwise, find the probability that the player
   1. progresses to the next stage,[3]
   2. completes exactly \(15\) tasks in the first set of tasks given he does not progress to the next stage.[2]