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

1. A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac { 1 } { 3 }\) that it lands on blue. The spinner is spun repeatedly.

The random variable \(B\) represents the number of the spin when the spinner first lands on blue.

1. Find (i) \(\mathrm { P } ( B = 4 )\) (ii) \(\mathrm { P } ( B \leqslant 5 )\)
2. Find \(\mathrm { E } \left( B ^ { 2 } \right)\) Steve invites Tamara to play a game with this spinner.
   Tamara must choose a colour, either red or blue.
   Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(\mathrm { e } ^ { X }\) If Tamara chooses blue, her score is \(X ^ { 2 }\)
3. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses.

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

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.

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

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]

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(\(N \leqslant n\)) is greater than \(0.9\). [3]

An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20. Find the probability that she requires at least 6 shots to hit the bull's-eye. [3] When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm{P}(Y = r) = \frac{1}{2}(r - 1)(r - 2)\left(\frac{3}{20}\right)^3\left(\frac{17}{20}\right)^{r-3}.$$ [3] Simplify \(\frac{\mathrm{P}(Y = r + 1)}{\mathrm{P}(Y = r)}\), and hence find the set of values of \(r\) for which \(\mathrm{P}(Y = r + 1) < \mathrm{P}(Y = r)\). Deduce the most probable value of \(Y\). [4]

The discrete random variable \(X\) has a geometric distribution with mean 4. Find

1. P\((X = 5)\), [3]
2. P\((X > 5)\), [2]
3. the least integer \(N\) such that P\((X \leqslant N) > 0.9995\). [2]

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 \(X\). Show that the probability that a head is obtained when the coin is tossed once is \(\frac{2}{3}\). [2] Find

1. P(\(X = 4\)), [1]
2. P(\(X > 4\)), [2]
3. the least integer \(N\) such that P(\(X \leq N\)) \(> 0.999\). [3]

Lan starts a new job on Monday. He will catch the bus to work every day from Monday to Friday inclusive. The probability that he will get a seat on the bus has the constant value \(p\). The random variable \(X\) denotes the number of days that Lan will catch the bus until he is able to get a seat. The probability that Lan will not get a seat on the Monday, Tuesday, Wednesday or Thursday of his first week is 0.4096.

1. Show that \(p = 0.2\). [2]
2. Find the probability that Lan first gets a seat on Monday of the second week in his new job. [2]
3. Find the least integer \(N\) such that \(\text{P}(X \leqslant N) > 0.9\), and identify the day and the week that corresponds to this value of \(N\). [4]

Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).

1. State two conditions necessary for \(X\) to have a geometric distribution. [2]
2. Assuming that \(X\) has the distribution Geo(0.3), find
   1. P\((X = 5)\), [2]
   2. P\((X > 5)\). [3]
3. Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context. [2]

Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1.

1. Find the probability that
   1. the first time she succeeds is on her 5th attempt, [2]
   2. the first time she succeeds is after her 5th attempt, [2]
   3. the second time she succeeds is before her 4th attempt. [4]
   Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2. Sandra and Jill take turns attempting to hit the target, with Sandra going first.
2. Find the probability that the first person to hit the target is Sandra, on her
   1. 2nd attempt, [2]
   2. 10th attempt. [3]

A game is played with a token on a board with a grid printed on it. The token starts at the point \((0, 0)\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8, or 1 unit in the positive \(y\)-direction with probability 0.2. The token stops when it reaches a point with a \(y\)-coordinate of 1. It is given that the token stops at \((X, 1)\).

1. 1. Find the probability that \(X = 10\). [2]
   2. Find the probability that \(X < 10\). [3]
2. Find the expected number of steps taken by the token. [2]
3. Hence, write down the value of 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]
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive. [2]
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]