5.02h Geometric: mean 1/p and variance (1-p)/p^2

5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.

1. Find 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 6 throws are required to obtain a pair of tails.
   On a different occasion, a pair of fair coins is thrown 80 times.
4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.

1 Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the value of \(\mathrm { E } ( X )\).
2. Find the probability that exactly 5 throws are required to obtain a pair of heads.
3. Find the probability that fewer than 7 throws are required to obtain a pair of heads.

5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
[0pt]

1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.

2 Sam is a member of a soccer club. She is practising scoring goals. The probability that Sam will score a goal on any attempt is 0.7 , independently of all other attempts.

1. Sam makes 10 attempts at scoring goals. Find the probability that Sam will score goals on fewer than 8 of these attempts.
2. Find the probability that Sam's first successful attempt will be before her 5th attempt.
3. Wei is a member of the same soccer club. He is also practising scoring goals. The probability that Wei will score a goal on any attempt is 0.6 , independently of all other attempts. Wei is going to keep making attempts until he scores 3 goals.
   Find the probability that he scores his third goal on his 7th attempt.

3 Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25 .

1. Find the probability that Kayla takes more than 6 throws to achieve a success.
2. Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes.

1 A fair six-sided die, with faces marked \(1,2,3,4,5,6\), is thrown repeatedly until a 4 is obtained.

1. Find the probability that obtaining a 4 requires fewer than 6 throws.
   On another occasion, the die is thrown 10 times.
2. Find the probability that a 4 is obtained at least 3 times.

2 An ordinary fair die is thrown until a 6 is obtained.

1. Find the probability that obtaining a 6 takes more than 8 throws.
   Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable \(X\).
2. Find the expected value of \(X\).
3. Find the probability that obtaining a pair of 6s takes either 10 or 11 throws.

3 Three fair 6-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time repeatedly. The score on each throw is the sum of the numbers on the uppermost faces.

1. Find the probability that a score of 17 or more is first obtained on the 6th throw.
2. Find the probability that a score of 17 or more is obtained in fewer than 8 throws.

4 Three fair 4-sided spinners each have sides labelled 1,2,3,4. The spinners are spun at the same time and the number on the side on which each spinner lands is recorded. The random variable \(X\) denotes the highest number recorded.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 64 }\).
2. Complete the probability distribution table for \(X\).

   \(x\)	1	2	3	4
   \(\mathrm { P } ( X = x )\)		\(\frac { 7 } { 64 }\)	\(\frac { 19 } { 64 }\)

   On another occasion, one of the fair 4 -sided spinners is spun repeatedly until a 3 is obtained. The random variable \(Y\) is the number of spins required to obtain a 3 .
3. Find \(\mathrm { P } ( Y = 6 )\).
4. Find \(\mathrm { P } ( Y > 4 )\).

2 Hazeem repeatedly throws two ordinary fair 6-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains.

1. Find the probability that it takes exactly 5 throws of the two dice for Hazeem to obtain a score of 8 or more.
2. Find the probability that it takes no more than 4 throws of the two dice for Hazeem to obtain a score of 8 or more.
3. For 8 randomly chosen throws of the two dice, find the probability that Hazeem obtains a score of 8 or more on fewer than 3 occasions.

2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.

1. Find the probability that it takes fewer than 7 spins for George to obtain a 5 .
   George spins the spinner 10 times.
2. Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.

5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.

1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
   2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
2. Find the probability that at least three of them will pass tomorrow.
   75 people will take their advanced driving test next week.
   [0pt]
3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]

1 Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.

1. Find \(\mathrm { P } ( X < 8 )\).
2. Find the probability that Nicola obtains a 6 for the second time on her 8th throw. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-02_2717_35_109_2012}

7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.

1. Find \(\mathrm { P } ( X = 5 )\) .
   ............................................................................................................................................
2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
3. Find the probability that Leno scores his second goal on or before his 5th attempt. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017} \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 The random variable \(X\) has the geometric distribution \(\operatorname { Geo } ( p )\).

1. Show that the probability generating function of \(X\) is \(\frac { \mathrm { pt } } { 1 - \mathrm { qt } }\), where \(\mathrm { q } = 1 - \mathrm { p }\).
2. Use the probability generating function of \(X\) to show that \(\operatorname { Var } ( X ) = \frac { \mathrm { q } } { \mathrm { p } ^ { 2 } }\).
   Kenny throws an ordinary fair 6-sided dice repeatedly. The random variable \(X\) is the number of throws that Kenny takes in order to obtain a 6 . The random variable \(Z\) denotes the sum of two independent values of \(X\).
3. Find the probability generating function of \(Z\).

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
   1. the sixth person selected,
   2. not among the first 10 people selected.
   3. Find the expected number of people selected.

6 A coin is biased so that the probability that it will show heads on any throw is \(\frac { 2 } { 3 }\). The coin is thrown repeatedly. The number of throws up to and including the first head is denoted by \(X\). Find

1. \(\mathrm { P } ( X = 4 )\),
2. \(\mathrm { P } ( X < 4 )\),
3. \(\mathrm { E } ( X )\).

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

2 Independent trials, on each of which the probability of a 'success' is \(p ( 0 < p < 1 )\), are being carried out. The random variable \(X\) counts the number of trials up to and including that on which the first success is obtained. The random variable \(Y\) counts the number of trials up to and including that on which the \(n\)th success is obtained.

1. Write down an expression for \(\mathrm { P } ( X = x )\) for \(x = 1,2 , \ldots\). Show that the probability generating function of \(X\) is $$\mathrm { G } ( t ) = p t ( 1 - q t ) ^ { - 1 }$$ where \(q = 1 - p\), and hence that the mean and variance of \(X\) are $$\mu = \frac { 1 } { p } \quad \text { and } \quad \sigma ^ { 2 } = \frac { q } { p ^ { 2 } }$$ respectively.
2. Explain why the random variable \(Y\) can be written as $$Y = X _ { 1 } + X _ { 2 } + \ldots + X _ { n }$$ where the \(X _ { i }\) are independent random variables each distributed as \(X\). Hence write down the probability generating function, the mean and the variance of \(Y\).
3. State an approximation to the distribution of \(Y\) for large \(n\).
4. The aeroplane used on a certain flight seats 140 passengers. The airline seeks to fill the plane, but its experience is that not all the passengers who buy tickets will turn up for the flight. It uses the random variable \(Y\) to model the situation, with \(p = 0.8\) as the probability that a passenger turns up. Find the probability that it needs to sell at least 160 tickets to get 140 passengers who turn up. Suggest a reason why the model might not be appropriate.

2 The random variable \(X ( X = 1,2,3,4,5,6 )\) denotes the score when a fair six-sided die is rolled.

1. Write down the mean of \(X\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\).
2. Show that \(\mathrm { G } ( t )\), the probability generating function (pgf) of \(X\), is given by $$\mathrm { G } ( t ) = \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) }$$ The random variable \(N ( N = 0,1,2 , \ldots )\) denotes the number of heads obtained when an unbiased coin is tossed repeatedly until a tail is first obtained.
3. Show that \(\mathrm { P } ( N = r ) = \left( \frac { 1 } { 2 } \right) ^ { r + 1 }\) for \(r = 0,1,2 , \ldots\).
4. Hence show that \(\mathrm { H } ( t )\), the pgf of \(N\), is given by \(\mathrm { H } ( t ) = ( 2 - t ) ^ { - 1 }\).
5. Use \(\mathrm { H } ( t )\) to find the mean and variance of \(N\). A game consists of tossing an unbiased coin repeatedly until a tail is first obtained and, each time a head is obtained in this sequence of tosses, rolling a fair six-sided die. The die is not rolled on the first occasion that a tail is obtained and the game ends at that point. The random variable \(Q ( Q = 0,1,2 , \ldots )\) denotes the total score on all the rolls of the die. Thus, in the notation above, \(Q = X _ { 1 } + X _ { 2 } + \ldots + X _ { N }\) where the \(X _ { i }\) are independent random variables each distributed as \(X\), with \(Q = 0\) if \(N = 0\). The pgf of \(Q\) is denoted by \(\mathrm { K } ( t )\). The familiar result that the pgf of a sum of independent random variables is the product of their pgfs does not apply to \(\mathrm { K } ( t )\) because \(N\) is a random variable and not a fixed number; you should instead use without proof the result that \(\mathrm { K } ( t ) = \mathrm { H } ( \mathrm { G } ( t ) )\).
6. Show that \(\mathrm { K } ( t ) = 6 \left( 12 - t - t ^ { 2 } - \ldots - t ^ { 6 } \right) ^ { - 1 }\).
   [0pt] [Hint. \(\left. \left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + \ldots + t ^ { 5 } \right) .\right]\)
7. Use \(\mathrm { K } ( t )\) to find the mean and variance of \(Q\).
8. Using your results from parts (i), (v) and (vii), verify the result that (in the usual notation for means and variances) $$\sigma _ { Q } { } ^ { 2 } = \sigma _ { N } { } ^ { 2 } \mu _ { X } { } ^ { 2 } + \mu _ { N } \sigma _ { X } { } ^ { 2 } .$$

6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is \(\frac { 3 } { 5 }\). Andrew tosses all five coins. It is given that the probability generating function of \(X\), the number of heads obtained on the unbiased coins, is \(\mathrm { G } _ { X } ( t )\), where $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$

1. Find \(G _ { Y } ( \mathrm { t } )\), the probability generating function of \(Y\), the number of heads on the biased coins.
2. The random variable \(Z\) is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of \(Z\), giving your answer as a polynomial.
3. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
4. Write down the value of \(\mathrm { P } ( Z = 3 )\).

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.

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.

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 .