5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)

6 A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn.
The total number of counters selected, including the white counter, is denoted by \(X\).

1. In the case when \(w = 2\),
   1. write down the distribution of \(X\),
   2. find \(P ( 3 < X \leq 7 )\).
   3. In the case when \(\mathrm { E } ( X ) = 2\), determine the value of \(w\).
   4. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour.

1. The independent random variables \(W\) and \(X\) have the following distributions.

$$W \sim \operatorname { Po } ( 4 ) \quad X \sim \mathrm {~B} ( 3,0.8 )$$

1. Write down the value of the variance of \(W\)
2. Determine the mode of \(X\) Show your working clearly. One observation from each distribution is recorded as \(W _ { 1 }\) and \(X _ { 1 }\) respectively.
3. Find \(\mathrm { P } \left( W _ { 1 } = 2 \right.\) and \(\left. X _ { 1 } = 2 \right)\)
4. Find \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)

6. A random sample of size \(n\) is taken from the random variable \(X\), which has a continuous uniform distribution over the interval [ \(0 , a\) ], \(a > 0\) The sample mean is denoted by \(\bar { X }\)

1. Show that \(Y = 2 \bar { X }\) is an unbiased estimator of \(a\) The maximum value, \(M\), in the sample has probability density function $$f ( m ) = \left\{ \begin{array} { c c } \frac { n m ^ { n - 1 } } { a ^ { n } } & 0 \leqslant m \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$
2. Find E(M)
3. Show that \(\operatorname { Var } ( M ) = \frac { n a ^ { 2 } } { ( n + 2 ) ( n + 1 ) ^ { 2 } }\) The estimator \(S\) is defined by \(S = \frac { n + 1 } { n } M\) Given that \(n > 1\)
4. state which of \(Y\) or \(S\) is the better estimator for \(a\). Give a reason for your answer.

4 The probability that an expert darts player hits the bullseye on any throw is 0.12 , independently of any other throw. The player throws darts at the bullseye until she hits it.

1. Find the probability that the player has to throw exactly six darts.
2. Find the probability that the player has to throw more than six darts.
3. (A) Find the mean number of darts that the player has to throw.
   (B) Find the variance of the number of darts that the player has to throw. The player continues to throw more darts at the bullseye after she has hit it for the first time.
4. Find the probability that the player hits the bullseye at least twice in the first ten throws.
5. Find the probability that the player hits the bullseye for the second time on the tenth throw.

7 On average one in five packets of a breakfast cereal contains a voucher for a discount on the next packet bought. Whether or not a packet contains a voucher is independent of other packets, and can only be determined by opening the packet.

1. State the distribution of the number of packets that need to be opened in order to find one which contains a voucher.
2. Determine the probability that exactly 4 packets have to be opened in order to find one which contains a voucher.
3. Determine the probability that exactly 10 packets have to be opened in order to find two which contain a voucher.
4. I have \(n\) packets, and I open them one by one until I find a voucher or until all the packets are open. Given that the probability that I find a voucher is greater than 0.99 , determine the least possible value of \(n\).

2 A group of friends live by the sea. Each day they look out to sea in the hope of seeing a dolphin. The probability that they see a dolphin on any day is 0.15 . You should assume that this probability is not affected by whether or not they see a dolphin on any other day.

1. Explain why you can use a geometric distribution to model the number of days that it takes for them to first see a dolphin.
2. Find the probability that they see a dolphin for the first time on the fifth day.
3. Find the probability that they do not see a dolphin for at least 10 days.
4. Determine the mean and the variance of the number of days that it takes for them to see a dolphin.

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.

3 A child is trying to draw court cards from an ordinary pack of 52 cards (court cards are Kings, Queens and Jacks; there are 12 in a pack). She draws cards, one at a time, with replacement, from the pack. Find the probabilities of the following events.

1. She draws a court card for the first time on the sixth try.
2. She draws a court card at least once in the first six tries.
3. She draws a court card for the second time on the sixth try.
4. She draws at least two court cards in the first six tries.

2 A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.

1. Determine \(\mathrm { P } ( X = 4 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
   • The player has \(n\) attempts to score a goal.
     1. Determine the least value of \(n\) for which the probability that the player first scores a goal on the \(n\)th attempt is less than 0.001 .
     2. Determine the least value of \(n\) for which the probability that the player scores at least one goal in \(n\) attempts is at least 0.999.

5 In a recent report, it was stated that \(40 \%\) of working people have a degree. For the whole of this question, you should assume that this is true. A researcher wishes to interview a working person who has a degree. He asks working people at random whether they have a degree and counts the number of people he has to ask until he finds one with a degree.

1. Find the probability that he has to ask 5 people.
2. Find the mean number of people the researcher has to ask. Subsequently, the researcher decides to take a random sample from the population of working people.
3. A random sample of 5 working people is chosen. What is the probability that at least one of them has a degree?
4. How large a random sample of working people would the researcher need to take to ensure that the probability that at least one person has a degree is 0.99 or more?

2 A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that \(12 \%\) of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.

1. Find the probability that the fifth person the researcher asks is the first to have watched the programme.
2. Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
3. Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
4. Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.

4 Alex is practising bowling at a cricket wicket. Every time she bowls a ball, she has a \(30 \%\) chance of hitting the wicket.

1. Assuming that successive bowls are independent, determine the probability that Alex first hits the wicket on her third attempt.
2. Determine the probability that Alex hits the wicket for the fourth time on her tenth attempt.

1 A fair spinner has ten sectors, labelled \(1,2 , \ldots , 10\). In order to start a game, Kofi has to obtain an 8,9 or 10 on the spinner.

1. Find the probability that Kofi starts the game on the third spin.
2. Find the probability that Kofi takes at least 5 spins to start the game.
3. Determine the probability that the number of spins required to start the game is within 1 standard deviation of its mean.

1 When a footballer takes a penalty kick the result is that either a goal is scored or a goal is not scored. It is known that, on average, a certain footballer scores a goal on \(85 \%\) of penalty kicks. During one practice session, the footballer decides to take penalty kicks until a goal is not scored. You may assume that the outcome of each penalty kick that the footballer takes is independent of the outcome of each other penalty kick. The random variable representing the number of penalty kicks up to and including the first penalty kick that does not result in a goal is denoted by \(X\).

1. State one further assumption that is necessary for \(X\) to be modelled by a Geometric distribution. For the remainder of this question you may assume that this assumption is valid.
2. Find each of the following.

2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let \(X\) represent the number of dead pixels on a monitor screen of this type.

1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use a Poisson distribution to calculate estimates of each of the following probabilities.
   For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of \(n\) pixels. Use a binomial distribution to find the least value of \(n\) for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean \(\lambda\).
3. Given that the probability of finding at least one dead pixel is 0.8 , find \(\lambda\).

4 Cards are drawn at random from a standard pack of 52 cards, one at a time, until one of the 4 aces is drawn. After each card is drawn, it is replaced in the pack before the next one is drawn. The random variable \(X\) represents the number of draws required to draw the first ace.

1. State fully the distribution of \(X\).
2. Find \(\mathrm { P } ( X = 10 )\).
3. Find each of the following.
   A further \(k\) aces are added to the full pack and the process described above is repeated. The random variable \(Y\) represents the number of draws required to draw the first ace.
4. In this question you must show detailed reasoning. Given that \(\mathrm { P } ( Y = 2 ) = \frac { 8 } { 81 }\), find the two possible values of \(k\).

5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.

1. 1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
   2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
3. Determine \(\mathrm { P } ( X < 5 )\).

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)

7 Amir is trying to thread a needle. On each attempt the probability that he is successful is 0.3 , independently of any other attempt. The random variable \(X\) represents the number of attempts that he takes to thread the needle.

1. Find \(\mathrm { P } ( X = 5 )\).
2. During the course of a day, Amir has to thread 6 needles. Determine the probability that it takes him more than 3 attempts to be successful for at least 4 of the 6 needles.
3. Amaya is also trying to thread a needle. On each attempt the probability that she is successful is \(p\), independently of any other attempt. The probability that Amaya takes 2 attempts to thread a particular needle is \(\frac { 28 } { 121 }\). Determine the possible values of \(p\).

3 A tennis player is practising her serve. Each time she serves, she has a \(55 \%\) chance of being successful (getting the serve in the required area without hitting the net). You should assume that whether she is successful on any serve is independent of whether she is successful on any other serve.

1. Find the probability that the player is not successful in any of her first three serves.
2. Determine the probability that the player is successful at least 10 times in her first 20 serves.
3. Determine the probability that the player is successful for the first time on her fifth serve.
4. Determine the probability that the player is successful for the fifth time on her tenth serve. Another player is also practising his serve. Each time he serves, he has a probability \(p\) of being successful. You should assume that whether he is successful on any serve is independent of whether he is successful on any other serve. The probability that he is successful for the first time on his second serve is 0.2496 and the probability that he is successful on both of his first two serves is less than 0.25 .
5. Determine the value of \(p\).

2 On average 1 in 4000 people have a particular antigen in their blood (an antigen is a molecule which may cause an adverse reaction). \begin{enumerate}[label=(\alph*)] \item

1. A random sample of 1200 people is selected. The random variable \(X\) represents the number of people in the sample who have this antigen in their blood. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use either a binomial or a Poisson distribution to calculate each of the following probabilities.

3 In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

1. Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
2. Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
3. Find the probability that in a sequence of 20 messages, there are no misunderstood messages.
4. Determine the expected number of messages required for 3 of them to be misunderstood.
5. Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.

1. A chocolate manufacturer places special tokens in \(2 \%\) of the bars it produces so that each bar contains at most one token. Anyone who collects 3 of these tokens can claim a prize.

Andreia buys a box of 40 bars of the chocolate.

1. Find the probability that Andreia can claim a prize. Barney intends to buy bars of the chocolate, one at a time, until he can claim a prize.
2. Find the probability that Barney can claim a prize when he buys his 40th bar of chocolate.
3. Find the expected number of bars that Barney must buy to claim a prize.

1. A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac { 1 } { 3 }\) that it lands on blue. The spinner is spun repeatedly.

The random variable \(B\) represents the number of the spin when the spinner first lands on blue.

1. Find (i) \(\mathrm { P } ( B = 4 )\) (ii) \(\mathrm { P } ( B \leqslant 5 )\)
2. Find \(\mathrm { E } \left( B ^ { 2 } \right)\) Steve invites Tamara to play a game with this spinner.
   Tamara must choose a colour, either red or blue.
   Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(\mathrm { e } ^ { X }\) If Tamara chooses blue, her score is \(X ^ { 2 }\)
3. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses.