State assumptions for geometric model

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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 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 \(k\).

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.

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.

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 \(X\) th 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.

1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by \(X\).

1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X \sim \operatorname { Geo } ( 0.4 )\).
2. Find \(\mathrm { P } ( X < 6 )\).
3. Find \(\mathrm { E } ( X )\).
4. Find \(\operatorname { Var } ( X )\).