Mean/expectation of geometric distribution

1 An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable \(X\).

1. Write down the mean of \(X\).
2. Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw.
3. Find the probability that a 5 is first obtained in fewer than 10 throws.

5 On average 1 in 20 members of the population of this country has a particular DNA feature. Members of the population are selected at random until one is found who has this feature.

1. Find the probability that the first person to have this feature is
   (a) the sixth person selected,
   (b) not among the first 10 people selected.
2. Find the expected number of people selected.

2 A random variable \(T\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find

1. \(\mathrm { P } ( T = 4 )\),
2. \(\mathrm { P } ( T > 4 )\),
3. \(\mathrm { E } ( T )\).

2 The probability that a certain sample of radioactive material emits an alpha-particle in one unit of time is 0.14 . In one unit of time no more than one alpha-particle can be emitted. The number of units of time up to and including the first in which an alpha-particle is emitted is denoted by \(T\).

1. Find the value of
   (a) \(\mathrm { P } ( T = 5 )\),
   (b) \(\mathrm { P } ( T < 8 )\).
2. State the value of \(\mathrm { E } ( T )\).

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1. A random variable \(X\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find
   (a) \(\mathrm { E } ( \mathrm { X } )\),
   (b) \(\mathrm { P } ( \mathrm { X } = 4 )\),
   (c) \(P ( X > 4 )\).
2. A random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\), and \(q = 1 - p\).
   (a) Show that \(P ( Y\) is odd \() = p + q ^ { 2 } p + q ^ { 4 } p + \ldots\).
   (b) Use the formula for the sum to infinity of a geometric progression to show that $$P ( Y \text { is odd } ) = \frac { 1 } { 1 + q }$$ \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
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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.

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

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 \(X\).

4. 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 \(X\).
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2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.
From tomorrow, I will count the number, \(X\), of nights on which I look for shooting stars, up to and including the first successful night. Find \(\mathrm { E } ( X )\).

1 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 \(X\).