5.02f Geometric distribution: conditions

4 Each packet of the breakfast cereal Fizz contains one plastic toy animal. There are five different animals in the set, and the cereal manufacturers use equal numbers of each. Without opening a packet it is impossible to tell which animal it contains. A family has already collected four different animals at the start of a year and they now need to collect an elephant to complete their set. The family is interested in how many packets they will need to buy before they complete their set.

1. Name an appropriate distribution with which to model this situation. State the value(s) of any parameter(s) of the distribution, and state also any assumption(s) needed for the distribution to be a valid model.
2. Find the probability that the family will complete their set with the third packet they buy after the start of the year.
3. Find the probability that, in order to complete their collection, the family will need to buy more than 4 packets after the start of the year.

6 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that

1. Malik throws a 6 for the first time on his third attempt,
2. Malik needs at most ten attempts to throw a 6.

4 The random variable \(U\) has the distribution \(\operatorname { Geo } ( p )\).

1. Show, from the definition, that the probability generating function ( pgf ) of \(U\) is given by $$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$ where \(q = 1 - p\).
2. Explain why the condition \(| t | < \frac { 1 } { q }\) is necessary.
3. Use the pgf to obtain \(\mathrm { E } ( U )\). Each packet of Corn Crisp cereal contains a voucher and \(20 \%\) of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let \(X\) denote the number of packets bought by a family up to and including the one from which the \(4 ^ { \text {th } }\) gold star is obtained.
4. Obtain the pgf of \(X\).
5. Find \(\mathrm { P } ( X = 6 )\).

11

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).

3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is \(\frac { 1 } { 8 }\). It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.

1. Calculate the probability that Erika first sees a woodpecker
   1. on the third day,
   2. after the third day.
   3. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
   4. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.

2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find

1. \(\mathrm { P } ( X = 3 )\),
2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
4. Find the probability that the total of these two values is 3 .

6 The diagrams illustrate all or part of the probability distributions of the discrete random variables \(V , W , X , Y\) and \(Z\). \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_365_370_296} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_376_370_838} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_362_370_1400} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_421_359_879_580} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_355_881_1142}

1. One of these variables has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } \right)\). State, with a reason, which variable this is.
2. One of these variables has the distribution \(\mathrm { B } \left( 4 , \frac { 1 } { 2 } \right)\). State, with reasons, which variable this is. \(760 \%\) of the voters at a certain polling station are women. Voters enter the polling station one at a time. The number of voters who enter, up to and including the first woman, is denoted by \(X\).

9

1. A clock is designed to chime once each hour, on the hour. The clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 10 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day. Find the probability that the first time it does not chime is
   1. at 0600 on that day,
   2. before 0600 on that day.
   3. Another clock is designed to chime twice each hour: on the hour and at 30 minutes past the hour. This clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 20 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day.
      (a) Find the probability that the first time it does not chime is at either 0030 or 0130 on that day.
      (b) Use the formula for the sum to infinity of a geometric progression to find the probability that the first time it does not chime is at 30 minutes past some hour.

9 Each day Harry makes repeated attempts to light his gas fire. If the fire lights he makes no more attempts. On each attempt, the probability that the fire will light is 0.3 independent of all other attempts. Find the probability that

1. the fire lights on the 5th attempt,
2. Harry needs more than 1 attempt but fewer than 5 attempts to light the fire. If the fire does not light on the 6th attempt, Harry stops and the fire remains unlit.
3. Find the probability that, on a particular day, the fire lights.
4. Harry's week starts on Monday. Find the probability that, during a certain week, the first day on which the fire lights is Wednesday.

5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .

1. Find the probability that the first time Jack wins a ticket is
   1. on his 8th attempt,
   2. after his 8th attempt.
   3. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
   4. Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.

8 A game is played with a fair, six-sided die which has 4 red faces and 2 blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or until the die has been thrown 4 times.

1. In the answer book, complete the probability tree diagram for one turn. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-5_314_302_1000_884}
2. Find the probability that in one particular turn the die is thrown 4 times.
3. Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more times than Beryl.
4. State one change that needs to be made to the rules so that the number of throws in one turn will have a geometric distribution.

5 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that

1. Malik throws a 6 for the first time on his third attempt,
2. Malik needs at most ten attempts to throw a 6.

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 \(Y = \frac { 1 } { T - 1 }\).

Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws, \(x\), needed is recorded and the results are summarised in the following table. State a type of distribution that could be used to fit the data given in the table above. Fit a distribution of this type in which the probability of throwing an even number for each die is 0.6 and carry out a goodness of fit test at the 5\% significance level. For each of these dice, it is known that the probability of obtaining a 6 when it is thrown is 0.25 . Ten of these dice are each thrown 5 times. Find the probability that at least one 6 is obtained on exactly 4 of the 10 dice.

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 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.

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 \(a\).