A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The number of throws taken is denoted by the random variable \(X\).
(i) State the expected value of \(X\).

(ii) Find the probability that exactly 3 throws are required to obtain a pair of tails.

(iii) Find the probability that fewer than 4 throws are required to obtain a pair of tails.

(iv) Find the least integer \(N\) such that the probability of obtaining a pair of tails in fewer than \(N\) throws is more than 0.95 .

A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
(i) State the mean value of \(X\).

(ii) Find the probability that obtaining a 3 or a 4 takes exactly 6 throws.

(iii) Find the probability that obtaining a 3 or a 4 takes more than 4 throws.

(iv) Find the greatest integer \(n\) such that the probability of obtaining a 3 or a 4 in fewer than \(n\) throws is less than 0.95.

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 .
(i) Show that \(p=0.2\).

(ii) Find the probability that Lan first gets a seat on Monday of the second week in his new job.

(iii) Find the least integer \(N\) such that \(\mathrm{P}(X \leqslant N)>0.9\), and identify the day and the week that corresponds to this value of \(N\).

A fair die is thrown repeatedly until a 6 is obtained.
(i) Find the probability that obtaining a 6 takes no more than four throws.
(ii) Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\) th throw is more than 0.95 .

A pair of fair dice is thrown repeatedly until a pair of sixes is obtained. The number of throws taken is denoted by the random variable \(X\).
(i) Find the mean value of \(X\).

(ii) Find the probability that exactly 12 throws are required to obtain a pair of sixes.

(iii) Find the probability that more than 12 throws are required to obtain a pair of sixes.

At a ski resort, the probability of snow on any particular day is constant and equal to \(p\). The skiing season begins on 1 November. The random variable \(X\) denotes the day of the skiing season on which the first snowfall occurs. (For example, if the first snowfall is on 5 November, then \(X=5\).) The variance of \(X\) is \(\frac{4}{9}\).
(i) Show that \(4 p^{2}+9 p-9=0\) and hence find the value of \(p\).

(ii) Find the probability that the first snowfall will be on 3 November.
(iii) Find the probability that the first snowfall will not be before 4 November.
(iv) Find the least integer \(N\) so that the probability of the first snowfall being on or before the \(N\) th day of November is more than 0.999 .

The probability that a driver passes an advanced driving test has a fixed value \(p\) for each attempt. A driver keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for the driver to pass. The variance of \(X\) is 3.75 .
(i) Show that \(15 p^{2}+4 p-4=0\) and hence find the value of \(p\).

(ii) Find \(\mathrm{P}(X=5)\).

(iii) Find \(\mathrm{P}(3 \leqslant X \leqslant 7)\).

\(6 Y\) is a discrete random variable which takes the values \(0,2,4, \ldots\) The probability generating function of \(Y\) is given by
\(\mathrm{G}_{Y}(t)=\frac{k}{1-a t^{2}}\)
(a) Find \(k\) in terms of \(a\).

(b) Show that \(\mathrm{P}(Y\gt 2)=a^{2}\).

It is now given that \(a=0.2\).
(c) Find the value of \(\mathrm{E}(Y)\).

A bag contains 7 red balls and 3 blue balls. Kieran selects 2 balls at random, without replacement. The number of red balls selected by Kieran is denoted by \(X\), and the number of different colours present in Kieran's selection is denoted by \(Y\).
(a) Find the probability generating functions, \(\mathrm{G}_{X}(t)\) of \(X\) and \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the sum of the number of red balls and the number of different colours present in Kieran's selection. Kieran claims that the probability generating function of \(Z\) is equal to \(\mathrm{G}_{X}(t) \times \mathrm{G}_{Y}(t)\).
(b) Explain why Kieran is incorrect.

(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).

(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).

Harry has three coins.
- One coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac{1}{3}\).
- The second coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac{1}{4}\).
- The third coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac{1}{5}\).

The random variable \(X\) is the number of heads that Harry obtains when he throws all three coins together.
(a) Find the probability generating function of \(X\).

Isaac has two fair coins. The random variable \(Y\) is the number of heads that Isaac obtains when he throws both of his coins together. The random variable \(Z\) is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
(c) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).

The random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=k\left(1+3 t+4 t^{2}\right)\)
where \(k\) is a constant.
(a) Show that \(\mathrm{E}(X)=\frac{11}{8}\).

The random variable \(Y\) has probability generating function \(\mathrm{G}_{Y}(t)\) given by
\(\mathrm{G}_{Y}(t)=\frac{1}{3} t^{2}(1+2 t)\)

The random variables \(X\) and \(Y\) are independent and \(Z=X+Y\).
(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).

(c) Use your answer to part (b) to find the value of \(\operatorname{Var}(Z)\).

(d) Write down the most probable value of \(Z\).

Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is \(\frac{2}{3}\). Jason throws all three coins. The number of heads obtained is denoted by \(X\).

(a) Find the probability generating function \(G_X(t)\) of \(X\).

Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by \(Y\). It is given that \(G_Y(t)=\frac14+\frac12t+\frac14t^2\). The random variable \(Z\) is the total number of heads obtained when Jason throws all five coins.

(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(c) Find \(E(Z)\).

Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag, without replacement. The number of red balls selected by Tanji is denoted by \(X\).
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is \(\frac{1}{4}\). He throws the two coins at the same time. The number of heads obtained is denoted by \(Y\).
(b) Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the sum of the number of red balls selected by Tanji and the number of heads obtained.
(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).

\(X\) is a discrete random variable which takes the values \(0,2,4, \ldots\). The probability generating function of \(X\) is given by
\(\mathrm{G}_{X}(t)=\frac{1}{3-2 t^{2}} .\)
(a) Find \(\mathrm{E}(X)\) and \(\operatorname{Var}(X)\).

(b) Find \(\mathrm{P}(X=4)\).

Nine balls labelled \(1,2,3,4,5,6,7,8,9\) are placed in a bag. Kai selects three balls at random from the bag, without replacement. The random variable \(X\) is the number of balls selected by Kai that are labelled with a multiple of 3 .
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

The balls are replaced in the bag.
Jacob now selects two balls at random from the bag, without replacement. The random variable \(Y\) is the number of balls selected by Jacob that are labelled with an even number.
(b) Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the sum of the number of balls that are labelled with a multiple of 3 selected by Kai and the number of balls that are labelled with an even number selected by Jacob.
(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).

The random variable \(X\) is such that \(\mathrm{P}(X=r)=k r^{2}\) for \(r=1,2,3,4\), where \(k\) is a constant.
(a) Find the value of \(k\).

(b) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

The random variable \(Y\) has probability generating function \(\mathrm{G}_{Y}(t)=\frac{1}{4}+\frac{1}{2} t+\frac{1}{4} t^{2}\).
The random variable \(Z\) is the sum of \(X\) and \(Y\).
(c) Assuming that \(X\) and \(Y\) are independent, find the probability generating function \(\mathrm{G}_{Z}(t)\) of \(Z\) as a polynomial in \(t\).
(d) Given that \(\mathrm{E}(Z)=\frac{13}{3}\), use \(\mathrm{G}_{Z}(t)\) to find \(\operatorname{Var}(Z)\).

A bag contains 4 red balls and 6 blue balls. Rassa selects two balls at random, without replacement, from the bag. The number of red balls selected by Rassa is denoted by \(X\).
(a) Find the probability generating function, \(\mathrm{G}_{X}(t)\), of \(X\).

Rassa also tosses two coins. One coin is biased so that the probability of a head is \(\frac{2}{3}\). The other coin is biased so that the probability of a head is \(p\). The probability generating function of \(Y\), the number of heads obtained by Rassa, is \(\mathrm{G}_{Y}(t)\). The coefficient of \(t\) in \(\mathrm{G}_{Y}(t)\) is \(\frac{7}{12}\).
(b) Find \(\mathrm{G}_{Y}(t)\).

The random variable \(Z\) is the sum of the number of red balls selected and the number of heads obtained by Rassa.
(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial.
(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).

Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by \(X\).
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Hassan has three coins, two of which are biased so that the probability of obtaining a head when the coin is tossed is \(\frac{1}{3}\). The corresponding probability for the third coin is \(\frac{1}{4}\). The number of heads obtained by Hassan when he tosses these three coins is denoted by \(Y\).
(b) Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the total number of heads obtained by Keira and Hassan.
(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).

(e) Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

The random variable \(X\) has the binomial distribution \(\mathrm{B}(n, p)\).
(a) Write down an expression for \(\mathrm{P}(X=r)\) and hence show that the probability generating function of \(X\) is \((q+p t)^{n}\), where \(q=1-p\).
(b) Use the probability generating function of \(X\) to prove that \(\mathrm{E}(X)=n p\) and \(\operatorname{Var}(X)=n p(1-p)\).

Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac{1}{3}\) and the third coin is biased so that the probability of obtaining a head is \(\frac{1}{5}\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
(a) Find the probability generating function of \(X\).

Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice.

The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial.
(c) Use your answer to part (b) to find \(\mathrm{E}(Z)\).

The random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=\frac{1}{5}+p t+q t^{2}\)
where \(p\) and \(q\) are constants.
(a) Given that \(\mathrm{E}(X)=1.1\), find the numerical value of \(\operatorname{Var}(X)\).

The random variable \(Y\) has probability generating function \(\mathrm{G}_{Y}(t)\) given by
\(\mathrm{G}_{Y}(t)=\frac{2}{3} t\left(1+\frac{1}{2} t^{2}\right) .\)

The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
(b) Find the probability generating function of \(Z\).

(c) Find \(\mathrm{P}(Z\gt 2)\).

(d) State the most probable value of \(Z\).

Eric has three identical coins, each of which is biased so that the probability of obtaining a head when it is thrown is \(\frac{1}{3}\). The random variable \(X\) is the number of heads obtained when Eric throws the three coins at the same time.
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Eric also has two fair 6 -sided dice with faces numbered 1 to 6 . The random variable \(Y\) is the number of sixes obtained when Eric throws the two dice at the same time. It is given that the probability generating function of \(Y\) is \(\frac{25}{36}+\frac{10}{36} t+\frac{1}{36} t^{2}\).

Eric throws the three coins and the two dice. The random variable \(Z\) is the sum of the number of heads obtained and the number of sixes obtained.
(b) Find the probability generating function \(\mathrm{G}_{Z}(t)\) of \(Z\), expressing your answer as a polynomial in \(t\).

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(c) Use \(\mathrm{G}_{Z}(t)\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).