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.

5 In an archery competition, competitors are allowed up to three attempts to hit the bulls-eye. No one who succeeds may try again. \(45 \%\) of those entering the competition hit the bulls-eye first time. For those who fail to hit it the first time, \(60 \%\) of those attempting it for the second time succeed in hitting it. For those who fail twice, only \(15 \%\) of those attempting it for the third time succeed in hitting it. By drawing a tree diagram, or otherwise,

1. find the probability that a randomly chosen competitor fails at all three attempts,
2. find the probability that a randomly chosen competitor fails at the first attempt but succeeds at either the second or third attempt,
3. find the probability that a randomly chosen competitor succeeds in hitting the bulls-eye,
4. find the probability that a randomly chosen competitor requires exactly two attempts given that the competitor is successful.

15 In order to be accepted on a university course, a student needs to pass three exams.
The probability that the student passes the first exam is \(\frac { 3 } { 4 }\).
For each of the second and third exams, the probability of passing the exam is

• the same as the probability of passing the preceding exam if the student passed the preceding exam,
• half of the probability of passing the preceding exam if the student failed the preceding exam.
  1. Draw a tree diagram to represent the above information.
  2. Find the probability that the student passes all three exams.
  3. Find the probability that the student passes at least two of the exams.
  4. Find the probability that the student passes the third exam given that exactly two of the three exams are passed.

The probability that it will rain on any given day is \(x\). If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4. If he does not wear a hat, the probability that he wears a scarf is 0.1. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36. Find the value of \(x\). [3]