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

4 It is known that \(8 \%\) of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.

1. Find the probability that the first person that he finds who uses this browser is
   (A) the third person selected,
   (B) the second or third person selected.
2. Find the probability that at least one of the first 20 people selected uses this browser.

The continuous random variable \(T\) has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 2 \\ \frac { 2 } { ( t - 1 ) ^ { 3 } } & t \geqslant 2 \end{cases}$$

1. Find the distribution function of \(T\), and find also \(\mathrm { P } ( T > 5 )\).
2. Consecutive independent observations of \(T\) are made until the first observation that exceeds 5 is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding 5. Find \(\mathrm { P } ( N > \mathrm { E } ( N ) )\).
3. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { T - 1 }\).

7 A fair die is thrown until a 6 appears for the first time. Assuming that the throws are independent, find

1. the probability that exactly 5 throws are needed,
2. the probability that fewer than 8 throws are needed,
3. the least integer \(n\) such that the probability of obtaining a 6 before the \(n\)th throw is at least 0.99 .

6 A pair of coins is thrown repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\). State the expected value of \(X\). Find the probability that

1. exactly 4 throws are required to obtain a pair of heads,
2. fewer than 6 throws are required to obtain a pair of heads.

6 The score when two fair dice are thrown is the sum of the two numbers on the upper faces. Two fair dice are thrown repeatedly until a score of 6 is obtained. The number of throws taken is denoted by the random variable \(X\). Find the mean of \(X\). Find the least integer \(N\) such that the probability of obtaining a score of 6 in fewer than \(N\) throws is more than 0.95 .

9 At a ski resort, the probability of snow on any particular day is constant and equal to \(p\). The skiing season begins on 1 November. The random variable \(X\) denotes the day of the skiing season on which the first snowfall occurs. (For example, if the first snowfall is on 5 November, then \(X = 5\).) The variance of \(X\) is \(\frac { 4 } { 9 }\).

1. Show that \(4 p ^ { 2 } + 9 p - 9 = 0\) and hence find the value of \(p\).
2. Find the probability that the first snowfall will be on 3 November.
3. Find the probability that the first snowfall will not be before 4 November.
4. Find the least integer \(N\) so that the probability of the first snowfall being on or before the \(N\) th day of November is more than 0.999 .

7 The probability that a driver passes an advanced driving test has a fixed value \(p\) for each attempt. A driver keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for the driver to pass. The variance of \(X\) is 3.75 .

1. Show that \(15 p ^ { 2 } + 4 p - 4 = 0\) and hence find the value of \(p\).
2. Find \(\mathrm { P } ( X = 5 )\).
3. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\).

6 A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the mean value of \(X\).
2. Find the probability that obtaining a 3 or a 4 takes exactly 6 throws.
3. Find the probability that obtaining a 3 or a 4 takes more than 4 throws.
4. Find the greatest integer \(n\) such that the probability of obtaining a 3 or a 4 in fewer than \(n\) throws is less than 0.95.

7 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the expected value of \(X\).
2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
3. Find the probability that fewer than 4 throws are required to obtain a pair of tails.
4. Find the least integer \(N\) such that the probability of obtaining a pair of tails in fewer than \(N\) throws is more than 0.95 .

6 In a skiing resort, for each day during the winter season, the probability that snow will fall on that day is 0.2 , independently of any other day. The first day of the winter season is 1 December. Find, for the winter season,

1. the probability that the first snow falls on 20 December,
2. the probability that the first snow falls before 5 December,
3. the earliest date in December such that the probability that the first snow falls on or before that date is at least 0.95 .

6 A fair die is thrown until a 5 or a 6 is obtained. The number of throws taken is denoted by the random variable \(X\). State the mean value of \(X\). Find the probability that obtaining a 5 or a 6 takes more than 8 throws. Find the least integer \(n\) such that the probability of obtaining a 5 or a 6 in fewer than \(n\) throws is more than 0.99.

6 A pair of fair dice is thrown repeatedly until a pair of sixes is obtained. The number of throws taken is denoted by the random variable \(X\).

1. Find the mean value of \(X\).
2. Find the probability that exactly 12 throws are required to obtain a pair of sixes.
3. Find the probability that more than 12 throws are required to obtain a pair of sixes.

6 A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\).

1. Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-11_69_1571_450_328}
2. Find \(\mathrm { P } ( X = 4 )\).
3. Find \(\mathrm { P } ( X > 4 )\).
4. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).

15

1. Show that \(\sum _ { r = 1 } ^ { \infty } 0.99 ^ { r - 1 } \times 0.01 = 1\). Kofi is a very good table tennis player. Layla is determined to beat him.
   Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time. \(X\) is the discrete random variable "the number of matches they play in total". Kofi models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = 0.99 ^ { \mathrm { r } - 1 } \times 0.01 \quad r = 1,2,3,4 , \ldots\) Kofi states that he is \(95 \%\) certain that Layla will not beat him within 6 years.
2. Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
3. Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
4. Explain how Layla's model takes into account the fact that she practises between matches, but Kofi's does not. Layla states that she is \(95 \%\) certain that she will beat Kofi within the first 6 matches.
5. Determine whether Layla's statement is consistent with her model.

1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the Xth weekday of September.

1. State an assumption needed for \(X\) to be well modelled by a geometric distribution.
2. Find \(\mathrm { P } ( X = 11 )\).
3. Find \(\mathrm { P } ( X \leqslant 8 )\).
4. Find \(\operatorname { Var } ( X )\).
5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.

6 A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by \(p\).

1. Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of \(p\), for the variance of the number of green beads Sasha selects. Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the \(X\) th bead that Freda selects.
2. Assume that \(p = 0.3\). Find
   1. \(\mathrm { P } ( X \geqslant 5 )\),
   2. \(\operatorname { Var } ( X )\).
3. In fact, on the basis of a large number of observations of \(X\), it is found that \(\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )\). Estimate the value of \(p\).

7 Each of three students, \(\mathrm { X } , \mathrm { Y }\) and Z , was given an identical pack of 48 cards, of which 12 cards were red and 36 were blue. They were each told to carry out a different experiment, as follows: Student X: Choose a card from the pack, at random, 20 times altogether, with replacement. Record how many times you obtain a red card. Student Y: Choose a card from the pack, at random, 20 times altogether, without replacement. Record how many times you obtain a red card. Student Z: Choose single cards from the pack at random, with replacement, until you obtain the first red card. Record how many cards you have chosen, including the first red card.

1. Find the probability that student Z has to choose more than 8 cards in order to obtain the first red card. Each student carries out their experiment 30 times. The frequencies of the results recorded by each student are shown in the following table, but not necessarily with the rows in the order \(\mathrm { X } , \mathrm { Y } , \mathrm { Z }\) :

   	Number recorded	0	1	2	3	4	5	6	7	8	\(\geqslant 9\)	Observed Mean	Observed Variance
   \multirow{3}{*}{Observed Frequencies}	Student 1	0	0	1	3	7	8	6	4	1	0	5.03	1.97
   	Student 2	0	8	5	4	2	3	3	2	1	2	4.03	11.57
   	Student 3	0	1	2	5	4	6	5	3	4	0	4.97	3.70

   \section*{(b) In this question you must show detailed reasoning.} Two other students make the following statements about the results. For each of the statements, explain whether you agree with the statement. Do not carry out any hypothesis tests, but in each case you should give two justifications for your answer.
   1. "The second row is a good match with the expected results for student Z ."
   2. "The third row is definitely student X 's results."

7 A town council is planning to introduce a new set of parking regulations. An interviewer contacts randomly chosen people in the town and asks them whether they are in favour of the proposal. The first person who is not in favour of the regulation is the \(R\) th person interviewed. It can be assumed that the probability that any randomly chosen person is not in favour of the proposal is a constant \(p\), and that \(p\) does not equal 0 or 1 . Assume first that \(\mathrm { E } ( R ) = 10\).

1. Determine \(\mathrm { P } ( R \geqslant 14 )\). Now, without the assumption that \(\mathrm { E } ( R ) = 10\), consider a general value of \(p\).
   It is given that \(\mathrm { P } ( R = 3 ) - 0.4 \times \mathrm { P } ( R = 2 ) - a \times \mathrm { P } ( R = 1 ) = 0\), where \(a\) is a positive constant.
2. Determine the range of possible values of \(a\).

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 \(\mathrm { E } ( X )\).