5.02f Geometric distribution: conditions

6 Anika walks along a street that contains parked cars. The number of cars that Anika passes, up to and including the first car that is white, is denoted by \(X\).

1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Geo } ( p )\), where \(0 < p < 1\).
2. For \(p = 0.1\), find \(\mathrm { P } ( X > 6 )\). The number of cars that Anika passes, up to but not including the first car that is white, is denoted by \(Y\).
3. For a general value of \(p\), determine a simplified expression for \(\mathrm { E } ( Y ) \div \operatorname { Var } ( Y )\), in terms of \(p\). Ben walks along a different street that also contains parked cars. The number of cars that Ben passes, up to and including the first white car on which the last digit of the number plate is even is denoted by \(Z\). It may be assumed that \(Z\) can be well modelled by the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } p \right)\), where \(p\) is the parameter of the distribution of \(X\). It is given that \(\mathrm { P } ( \mathrm { Z } = 3 ) = \mathrm { kP } ( \mathrm { X } = 3 )\), where \(k\) is a positive constant.
4. Determine the range of possible values of \(k\).

2 Every time a spinner is spun, the probability that it shows the number 4 is 0.2 , independently of all other spins.

1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required.
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive.
3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\).

5 The discrete random variable \(X\) has a geometric distribution. It is given that \(\operatorname { Var } ( X ) = 20\).
Determine \(\mathrm { P } ( X \geqslant 7 )\).

7 The random variable \(D\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\operatorname { Var } ( D ) = \frac { 40 } { 9 }\).
Determine

1. \(\operatorname { Var } ( 3 D + 5 )\),
2. \(\mathrm { E } ( 3 \mathrm { D } + 5 )\),
3. \(\mathrm { P } ( D > \mathrm { E } ( D ) )\).

1 A researcher wishes to find people who say that they support a specific plan. Each day the researcher interviews people at random, one after the other, until they find one person who says that they support this plan. The researcher does not then interview any more people that day. The total number of people interviewed on any one day is denoted by \(R\).

1. Assume that in fact \(1 \%\) of the population would say that they support the plan.
   1. State an appropriate distribution with which to model \(R\), giving the value(s) of any parameter(s).
   2. Find \(\mathrm { P } ( 50 < R \leqslant 150 )\). The researcher incorrectly believes that the variance of a random variable \(X\) with any discrete probability distribution is given by the formula \([ \mathrm { E } ( X ) ] ^ { 2 } - \mathrm { E } ( X )\).
2. Show that, for the type of distribution stated in part (a), they will obtain the correct value of the variance, regardless of the value(s) of the parameter(s).

1 A certain section of a library contains several thousand books. A lecturer is looking for a book that refers to a particular topic. The lecturer believes that one-twentieth of the books in that section of the library contain a reference to that topic. However, the lecturer does not know which books they might be, so the lecturer looks in each book in turn for a reference to the topic. The first book the lecturer finds that refers to the topic is the \(X\) th book in which the lecturer looks.

1. A student says, "There is a maximum value of \(X\) as there is only a finite number of books. So a geometric distribution cannot be a good model for \(X\)." Explain whether you agree with the student.
2. 1. State one modelling assumption (not involving the total number of books) needed for \(X\) to be modelled by a geometric distribution in this context.
   2. Suggest a reason why this assumption may not be valid in this context. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Geo } ( 0.05 )\).
3. The probability that the lecturer needs to look in no more than \(n\) books is greater than 0.9 . Find the smallest possible value of \(n\).
4. The lecturer needs to find four different books that refer to the topic. Find the probability that the lecturer wants to look in exactly 40 books.

3 In a large collection of coloured marbles of identical size, the proportion of green marbles is \(p\). One marble is chosen randomly, its colour is noted, and it is then replaced. This process is repeated until a green marble is chosen. The first green marble chosen is the \(X\) th marble chosen.

1. You are given that \(p = 0.3\).
   1. Find \(\mathrm { P } ( 5 \leqslant X \leqslant 10 )\).
   2. Determine the smallest value of \(n\) for which \(\mathrm { P } ( X = n ) < 0.1\).
2. You are given instead that \(\operatorname { Var } ( X ) = 42\). Determine the value of \(\mathrm { E } ( X )\).

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.

2. A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter \(\mu\). The number 2 is on \(40 \%\) of the counters and the number 5 is on \(25 \%\) of the counters. All the remaining counters have numbers greater than 5 on them. A random sample of 10 counters is taken from the bag.

1. State whether or not each of the following is a statistic
   1. \(S =\) the sum of the numbers on the counters in the sample,
   2. \(D =\) the difference between the highest number in the sample and \(\mu\),
   3. \(F =\) the number of counters in the sample with a number 5 on them. The random variable \(T\) represents the number of counters in a random sample of 10 with the number 2 on them.
2. Specify the sampling distribution of \(T\). The counters are selected one by one.
3. Find the probability that the third counter selected is the first counter with the number 2 on it.

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 \(k\).

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 \(p\).