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

1. Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.
Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

1. Find the probability that Suzanne selects the blue counter on her 4th selection.
2. Find the probability that the blue counter is first selected on or after Jon's third selection.
3. Find the mean and standard deviation of the number of selections made until the blue counter is selected.
4. Find the probability that Suzanne wins the game.

1. Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.
   Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

1. Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.
2. Find the probability that Asha gets a red counter for the second time on her 9th draw. The probability of Davinda getting a red counter on any one draw is \(p\). Davinda draws counters until she gets \(n\) red counters. The random variable \(D\) is the number of counters Davinda draws. Given that the mean and the standard deviation of \(D\) are 4400 and 660 respectively,
3. find the value of \(p\). Jerry believes that his bag contains a smaller proportion of red counters than Asha's bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable \(J\) to be the number of counters drawn up to and including the first red counter.
4. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. Jerry gets a red counter for the first time on his 34th draw.
5. Giving a reason for your answer, state whether or not there is evidence that Jerry's bag contains a smaller proportion of red counters than Asha's bag. Given that the probability of Jerry getting a red counter on any one draw is 0.011
6. show that the power of the test is 0.702 to 3 significant figures.

1. A random sample of 150 observations is taken from a geometric distribution with parameter 0.3

Estimate the probability that the mean of the sample is less than 3.45

1. There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die. Estimate the probability that the mean number of rolls for the class is less than 27.2

1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)

When more than one game is played the games are independent.
Sam plays 20 games.

1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
5. Find, in terms of \(r\), Tessa's expected profit.

1. The probability of Richard winning a prize in a game at the fair is 0.15

Richard plays a number of games.

1. Find the probability of Richard winning his second prize on his 8th game,
2. State two assumptions that have to be made, for the model used in part (a) to be valid. M ary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable \(G\) represents the total number of games M ary plays.
3. Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game.

1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that \(5 \%\) of letters receive a favourable reply.

1. Bill sends a letter to each of 40 potential sponsors. Assuming that the number \(N\) of favourable responses can be modelled by a binomial distribution, find the mean and variance of \(N\).
2. Gill sends one letter at a time to potential sponsors. \(L\) is the number of letters she sends, up to and including the first letter that receives a favourable response.
   1. State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
   2. Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a \(90 \%\) chance of receiving at least one favourable reply.

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 \(\operatorname { Var } ( X )\).

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

7 Sasha tends to forget his passwords. He investigates whether the number of attempts he needs to log on to a system with a password can be modelled by a geometric distribution. On 60 occasions he records the number of attempts he needs to log on, and the results are shown in the table.

1. Test at the \(1 \%\) significance level whether the results are consistent with the distribution Geo(0.4).
   [0pt]
2. Suggest which two probabilities should be changed, and in what way, to produce an improved model. (Numerical values are not required.) You should give a reason for your suggestion. [3]

5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.

1. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
2. Find the probability that the first wet day in October is 8 October.
3. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.

4 The discrete random variable \(Y\) has probability distribution The standard deviation of \(Y\) is \(s\) 4

1. Show that \(s = 10.53\) correct to two decimal places.
   [0pt] [4 marks]
   4
2. The median of \(Y\) is \(m\) Find \(\mathrm { P } ( Y > m - 1.5 s )\)

4 A fair six-sided dice is rolled repeatedly. Find the probability of the following events.

1. A five occurs for the first time on the fourth roll.
2. A five occurs at least once in the first four rolls.
3. A five occurs for the second time on the third roll.
4. At least two fives occur in the first three rolls. The dice is rolled repeatedly until a five occurs for the second time.
5. Find the expected number of rolls required for two fives to occur. Justify your answer.

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.

1. Determine the probability that Jeff wins the game
   i) with his first shot,
   ii) with his second shot.
2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.

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 \(X\).

2 A bag contains four black balls and one white ball. A man chooses a ball at random. If it is a black ball, he replaces it and chooses another at random. If he chooses the white ball, he stops.

1. Name the probability distribution which models this situation.
2. Calculate the probability that he will make exactly three attempts before he stops.
3. Calculate the probability that he will make fewer than three attempts before he stops.

2 Jill is collecting picture cards given away in packets of a particular brand of breakfast cereal. There are five different cards in the complete set. Each packet contains one card which is equally likely to be any of the five cards in the set.

1. Find the probability that Jill has a complete set of cards from the first five packets that she buys.
2. At some point Jill needs just one more card to complete the set. Let \(X\) be the random variable that represents the number of additional packets that Jill will need to buy in order to complete the set.
   1. Write down the distribution of \(X\).
   2. State the expected number of additional packets that Jill will need to buy.
   3. Find the probability that Jill will need to buy at least 3 additional packets in order to complete the set.

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

1. Show that \(\mathrm { P } ( X > n ) = q ^ { n }\), where \(q = 1 - p\). You are given that \(\mathrm { P } ( X \geqslant 4 ) = 0.216\).
2. Use the result given in part (i) to find the value of \(p\) and \(\mathrm { P } ( X \leqslant 8 )\).
3. Write down \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

11 During the 30-day month of April, the probability that it will rain on any given day is 0.25 .

1. Find the probability that the first rainy day in April is the 7th of April, explaining any modelling assumptions you have made.
2. Given that it does not rain on the first 6 days of April, find the probability that it first rains on the 10th of April.
3. The probability that it first rains on the \(n\)th of April and next on the ( \(n + 3\) )th of April is 0.02 , correct to 1 significant figure. Determine \(n\).
4. Determine the expected number of dry days in April, given that it first rains on the 8th of April.

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform disc, of mass \(4m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(OA\), \(OB\) and \(OC\), each of mass \(m\) and length \(2a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42ma^2\). [5] The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(AO\) making an angle of \(30°\) with the horizontal. Find the angular speed of the wheel when \(AO\) is horizontal. [3] When \(AO\) is horizontal the disc becomes detached from the wheel. Find the angle that \(AO\) makes with the horizontal when the wheel first comes to instantaneous rest. [6] **OR** The continuous random variable \(T\) has probability density function given by $$f(t) = \begin{cases} 0 & t < 2, \\ \frac{2}{(t-1)^3} & t \geqslant 2. \end{cases}$$

1. Find the distribution function of \(T\), and find also P\((T > 5)\). [3]
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 P\((N > E(N))\). [3]
3. Find the probability density function of \(Y\), where \(Y = \frac{1}{T-1}\). [8]

The probability that a particular type of light bulb is defective is 0.01. A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. [2] The first defective bulb is the \(N\)th to be tested. Write down the value of E\((N)\). [1] Find the least value of \(n\) such that P\((N < n)\) is greater than 0.9. [3]