Name geometric distribution and parameter

8 Henry makes repeated attempts to light his gas fire. He makes the modelling assumption that the probability that the fire will light on any attempt is \(\frac { 1 } { 3 }\). Let \(X\) be the number of attempts at lighting the fire, up to and including the successful attempt.

1. Name the distribution of \(X\), stating a further modelling assumption needed. In the rest of this question, you should use the distribution named in part (i).
2. Calculate
   (a) \(\mathrm { P } ( X = 4 )\),
   (b) \(\mathrm { P } ( X < 4 )\).
3. State the value of \(\mathrm { E } ( X )\).
4. Henry has to light the fire once a day, starting on March 1st. Calculate the probability that the first day on which fewer than 4 attempts are needed to light the fire is March 3rd.

1 A researcher wishes to find people who say that they support a specific plan. Each day the researcher interviews people at random, one after the other, until they find one person who says that they support this plan. The researcher does not then interview any more people that day. The total number of people interviewed on any one day is denoted by \(R\).

1. Assume that in fact \(1 \%\) of the population would say that they support the plan.
   1. State an appropriate distribution with which to model \(R\), giving the value(s) of any parameter(s).
   2. Find \(\mathrm { P } ( 50 < R \leqslant 150 )\). The researcher incorrectly believes that the variance of a random variable \(X\) with any discrete probability distribution is given by the formula \([ \mathrm { E } ( X ) ] ^ { 2 } - \mathrm { E } ( X )\).
2. Show that, for the type of distribution stated in part (a), they will obtain the correct value of the variance, regardless of the value(s) of the parameter(s).

7 On average one in five packets of a breakfast cereal contains a voucher for a discount on the next packet bought. Whether or not a packet contains a voucher is independent of other packets, and can only be determined by opening the packet.

1. State the distribution of the number of packets that need to be opened in order to find one which contains a voucher.
2. Determine the probability that exactly 4 packets have to be opened in order to find one which contains a voucher.
3. Determine the probability that exactly 10 packets have to be opened in order to find two which contain a voucher.
4. I have \(n\) packets, and I open them one by one until I find a voucher or until all the packets are open. Given that the probability that I find a voucher is greater than 0.99 , determine the least possible value of \(n\).

1 When a footballer takes a penalty kick the result is that either a goal is scored or a goal is not scored. It is known that, on average, a certain footballer scores a goal on \(85 \%\) of penalty kicks. During one practice session, the footballer decides to take penalty kicks until a goal is not scored. You may assume that the outcome of each penalty kick that the footballer takes is independent of the outcome of each other penalty kick. The random variable representing the number of penalty kicks up to and including the first penalty kick that does not result in a goal is denoted by \(X\).

1. State one further assumption that is necessary for \(X\) to be modelled by a Geometric distribution. For the remainder of this question you may assume that this assumption is valid.
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Find the probability that the footballer takes exactly 3 penalty kicks.
   • Find the probability that the footballer takes at least 5 penalty kicks.

4 Cards are drawn at random from a standard pack of 52 cards, one at a time, until one of the 4 aces is drawn. After each card is drawn, it is replaced in the pack before the next one is drawn. The random variable \(X\) represents the number of draws required to draw the first ace.

1. State fully the distribution of \(X\).
2. Find \(\mathrm { P } ( X = 10 )\).
3. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   A further \(k\) aces are added to the full pack and the process described above is repeated. The random variable \(Y\) represents the number of draws required to draw the first ace.
4. In this question you must show detailed reasoning. Given that \(\mathrm { P } ( Y = 2 ) = \frac { 8 } { 81 }\), find the two possible values of \(k\).

6 A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn. The total number of counters selected, including the white counter, is denoted by \(X\).

1. In the case when \(w = 2\),
   (a) write down the distribution of \(X\),
   (b) find \(P ( 3 < X \leq 7 )\).
2. In the case when \(\mathrm { E } ( X ) = 2\), determine the value of \(w\).
3. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour.