Conditional or compound probability scenarios

2 In a game of chance there are 32 slots, numbered 1 to 32, and on each turn a ball lands in one of them. You may assume that the process is completely random. You are given that \(X\) is the random variable denoting the number of the slot that the ball lands in on a given turn.

1. Suggest a suitable distribution to model \(X\). You should state the value(s) of any parameter(s).
2. Write down \(\mathrm { P } ( X = 7 )\). Players of the game start with a score of 0 . On each turn a player may choose to play the game by selecting a number. If the ball lands in the slot with that number then 15 is added to the player's score. Otherwise, the player's score is reduced by 1 . A player's score may become negative. A player decides to play the game, selecting the number 7 on each turn, until the ball lands in the slot numbered 7. You are given that \(Y\) is the random variable denoting the number of turns up to and including the turn in which the ball lands in the slot numbered 7.
3. Determine \(\mathrm { P } ( Y \leqslant 15 )\).
4. Determine the player's expected final score.

4 A pack of \(k\) cards is labelled \(1,2 , \ldots , k\). A card is drawn at random from the pack. The random variable \(X\) represents the number on the card.

1. Given that \(k > 10\), find \(\mathrm { P } ( X \geqslant 10 )\). You are now given that \(k = 20\).
2. A card is drawn at random from the pack and the number on it is noted. The card is then returned to the pack. This process is repeated until the second occasion on which the number noted is less than 9 . Find the probability that no more than 4 cards have to be drawn. Answer all the questions. Section B (95 marks)

4. Alyona, Dawn and Sergei are sometimes late for school. The events \(A , D\) and \(S\) are as follows:
A Alyona is late for school
D Dawn is late for school
S Sergei is late for school The Venn diagram below shows the three events \(A , D\) and \(S\) and the probabilities associated with each region of \(D\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(D\). \includegraphics[max width=\textwidth, alt={}, center]{b29b0411-8401-420b-9227-befe25c245d8-06_624_1068_845_479}

1. Write down 2 of the events \(A , D\) and \(S\) that are mutually exclusive. Give a reason for your answer. The probability that Sergei is late for school is 0.2 . The events \(A\) and \(D\) are independent.
2. Find the value of \(r\).
   (4) Dawn and Sergei's teacher believes that when Sergei is late for school, Dawn tends to be late for school.
3. State whether or not \(D\) and \(S\) are independent, giving a reason for your answer.
   (1)
4. Comment on the teacher's belief in the light of your answer to part (c).
   (1)
   (Total for Question 4 is 7 marks) \section*{Pearson Edexcel Level 3} \section*{GCE Mathematics} \section*{Paper 2: Mechanics}

   Specimen paper Time: \(\mathbf { 3 5 }\) minutes	Paper Reference(s)
   	\(\mathbf { 8 M A 0 } / \mathbf { 0 2 }\)
   You must have: Mathematical Formulae and Statistical Tables, calculator

   Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
   \section*{Information}
   \section*{Advice}