Conditional with three or more stages

6 Janice is playing a computer game. She has to complete level 1 and level 2 to finish the game. She is allowed at most two attempts at any level.

• For level 1 , the probability that Janice completes it at the first attempt is 0.6 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.3 .
• If Janice completes level 1, she immediately moves on to level 2.
• For level 2, the probability that Janice completes it at the first attempt is 0.4 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.2 .
  1. Show that the probability that Janice moves on to level 2 is 0.72 .
  2. Find the probability that Janice finishes the game.
  3. Find the probability that Janice fails exactly one attempt, given that she finishes the game.
     

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation \(T\) ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.

1. Find the probability that, when Deeti carries out operation \(T\), she takes a blue pen from her left pocket and then a blue pen from her right pocket. The random variable \(X\) is the number of blue pens in Deeti's left pocket after carrying out operation \(T\).
2. Find \(\mathrm { P } ( X = 1 )\).
3. Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.